#92908

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#95
commutator, prime radical	C₂³	C₈²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

159 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104);