#147774

order	512
nilpotency class	4
isoclinism	isoclinic class of #147773
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10991
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1626 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,15,20,33)(4,21,25,40)(5,13,7,11)(6,12)(8,14)(9,17)(10,24,37,27)(16,45)(18,44,51,62)(19,34)(22,54,52,61)(23,53)(26,41,28,38)(29,42)(30,59)(31,55,60,57)(32,48)(35,49)(36,56)(39,50)(43,58)(46,63)(47,64);(1,3,8,20)(2,9,14,34)(4,22,25,52)(5,19,7,17)(6,18,29,51)(10,35,37,63)(11,33,13,15)(12,39,42,32)(16,30,47,23)(21,45,40,64)(24,57,27,55)(26,50,28,48)(31,43,60,36)(38,44,41,62)(46,56,49,58)(53,61,59,54);(1,4)(2,10)(3,16)(5,23)(6,27)(7,30)(8,25)(9,31)(11,36)(12,40)(13,43)(14,37)(15,35)(17,22)(18,49)(19,52)(20,47)(21,42)(24,29)(26,58)(28,56)(32,61)(33,63)(34,60)(38,53)(39,54)(41,59)(44,64)(45,62)(46,51)(48,57)(50,55)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267961, #72267974, #72267975, …
maximal normal subgroup of	#72267961, #72267974, #72267975, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54970);