#92907

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=chi,e^a=ei,c^b=ch

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103);