#450795

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357991);