#433848

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 341044);