#444345

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351541);