#298569

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 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)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22)(23,30)(24,34)(25,29)(26,28)(27,33)(31,36)(32,35);(1,2,4,3)(7,14)(8,17)(11,16)(13,19)(21,23)(22,27)(24,32)(25,33)(26,31)(28,35)(29,30)(34,36);(5,7,8,15,9,14,17,20)(6,10,11,13,12,18,16,19)(23,31)(24,26)(27,35)(28,34)(30,36)(32,33);(1,3,4,2)(21,24,25,26)(22,28,29,34)(23,32,33,31)(27,35,30,36)

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205765);