#261302

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
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×#534	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1153 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)(7,12)(8,17)(9,14)(10,15)(11,16)(13,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(1,2,4,3)(5,7,11,17)(6,12,16,8)(9,13,10,20)(14,19,15,18)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(5,8,11,12)(6,13,16,20)(7,14,17,15)(9,18,10,19)(21,24,25,31)(22,27,28,35)(23,29,30,36)(26,32,33,34);(1,3,4,2)(5,9,11,10)(6,14,16,15)(7,13,17,20)(8,18,12,19)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168498);