#297369

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1102 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)(7,15)(8,17)(9,13)(10,19)(11,14)(12,16)(18,20)(21,22,25,28)(23,26,30,34)(24,27,32,35)(29,36,31,33);(1,2,4,3)(5,7,9,16)(6,12,13,15)(8,11,18,10)(14,17,19,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,34)(29,32)(35,36);(5,8,10,16,9,18,11,7)(6,12,14,17,13,15,19,20)(21,24)(22,27)(23,29,30,31)(25,32)(26,33,34,36)(28,35);(1,3,4,2)(6,13)(12,15)(14,19)(17,20)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,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,g^c=ghi

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, 204565);