#401692

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22,25,27)(23,30,32,36)(24,29,26,33)(28,34,35,31);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(22,27)(28,35)(29,33)(30,36);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(21,23,25,32)(22,28,27,35)(24,31,26,34)(29,36,33,30);(2,4)(21,24,25,26)(22,29,27,33)(23,31,32,34)(28,36,35,30)

Polycyclic presentation:

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

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, 308888);