#432137

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	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

1738 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=fi,d^2=fi,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, 339333);