#261202

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1068 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:

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

Polycyclic presentation:

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

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, 168398);