#92900

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#95
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16)(18,23,24,26)(20,28)(25,32,31,30)(27,29);(2,6)(3,9)(8,13)(10,15)(17,18,20,25,21,26,29,32,22,24,28,31,19,23,27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96);