#239581

order	512
nilpotency class	3
isoclinism	isoclinic class of #232373
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1001 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)(7,11)(8,13)(9,12)(10,14)(15,18)(16,20)(17,19)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(1,2)(3,4)(5,7,10,16)(6,11,14,20)(8,17,9,15)(12,18,13,19)(21,23)(22,26)(24,31)(25,30)(27,35)(28,33)(29,32)(34,36);(1,3,4,2)(6,12)(8,9)(11,18)(13,14)(15,17)(19,20)(24,32)(26,33)(27,36)(29,31);(5,8,10,9)(6,13,14,12)(7,15,16,17)(11,18,20,19)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,34,33,35)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=hi,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,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, 146777);