#341080

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248276);