#409808

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 317004);