#370485

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=fg,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277681);