#496180

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21738
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

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

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, 403376);