#349002

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
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	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256198);