#411225

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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)(3,6)(4,5)(7,8)(10,13)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26,30,32,29,27,28,31);(9,10,12,13)(11,15,16,14)(17,19,20,23)(18,21,22,24)(25,27,29,26)(28,31,30,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(26,27)(31,32);(1,3,4,7)(2,6,5,8)(18,22)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318421);