#347565

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
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	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254761);