#94001

order	512
nilpotency class	3
isoclinism	isoclinic class of #93290
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1274
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄×M₃₂	C₂²
socle	C₂	#7143
prime radical	C₂³	#349
LCS	G ▷ C₄² ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=ch,e^a=efh,f^a=fi,d^b=df,g^b=gi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1197);