#278702

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

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, 185898);