#297940

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1063 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205136);