#348995

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256191);