#498042

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22799
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405238);