#432134

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1510 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339330);