#359384

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,g^c=gi

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, 266580);