#343588

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

548 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)(5,7)(6,8)(10,13)(14,16)(17,18,22,24,20,23,21,26)(19,27,30,25,28,32,29,31);(1,3)(2,6)(4,7)(5,8)(10,14)(12,15)(13,16)(18,23)(19,28)(24,26)(29,30);(1,4,5,6)(2,3,7,8)(9,10)(11,13)(12,16)(14,15)(18,24)(21,22)(23,26)(25,27)(29,30)(31,32);(2,7)(3,8)(17,19,20,28)(18,25,23,31)(21,30,22,29)(24,27,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250784);