#492869

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#106
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂²×#340
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1581 conjugacy classes of subgroups; only the 864 normal subgroups are shown.

normal subgroups only

This group has 1581 conjugacy classes of subgroups — too many to draw. Only its 864 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,18,40)(4,10)(5,12)(6,27)(7,29)(8,14)(9,17,31,45)(11,25)(13,28)(16,39,41,59)(19,48,46,33)(20,49,47,32)(21,34)(22,36)(23,53)(24,54)(26,38)(30,42,55,60)(35,51)(37,52)(43,63,61,57)(44,64,62,56)(50,58);(1,3,6,19,28,47,26,45,8,18,25,46,7,20,5,17)(2,9,12,32,29,48,11,15,14,31,38,49,13,33,27,40)(4,16,23,43,52,62,50,60,21,41,51,61,24,44,22,42)(10,30,36,56,54,63,35,39,34,55,58,64,37,57,53,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,21)(9,30)(11,35)(12,36)(13,37)(14,34)(15,39)(17,42)(18,41)(19,43)(20,44)(25,51)(26,50)(27,53)(28,52)(29,54)(31,55)(32,56)(33,57)(38,58)(40,59)(45,60)(46,61)(47,62)(48,63)(49,64);(4,21)(10,34)(16,41)(22,50)(23,51)(24,52)(30,55)(35,53)(36,58)(37,54)(39,59)(42,60)(43,61)(44,62)(56,64)(57,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307063, #72307069, #72307131, …
maximal normal subgroup of	#72307063, #72307069, #72307131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400065);