#297523

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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

1130 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)(5,6,9,12)(7,10,14,18)(8,11,16,19)(13,20,15,17)(21,22,25,28)(23,30,31,27)(24,32,33,36)(26,35,34,29);(1,2,4,3)(5,7)(6,10)(8,15)(9,14)(11,17)(12,18)(13,16)(19,20)(21,23,25,31)(22,27,28,30)(24,26,33,34)(29,36,35,32);(5,8)(6,11)(7,13,14,15)(9,16)(10,17,18,20)(12,19)(21,24,26,31,25,33,34,23)(22,27,29,32,28,30,35,36);(1,3,4,2)(8,16)(11,19)(13,15)(17,20)(22,28)(27,30)(29,35)(32,36)

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,d^c=dhi,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, 204719);