#152166

order	512
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12731
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224985, #72225006, #72225010
maximal normal subgroup of	#72224985, #72225006, #72225010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59362);