#262520

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	#59652	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1017 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)(7,11)(8,13)(9,12)(10,14)(15,18)(16,20)(17,19)(21,22,26,25)(23,27,30,33)(24,28,32,34)(29,35,36,31);(1,2,4,3)(5,7,10,16)(6,11,14,20)(8,15,9,17)(12,19,13,18)(21,23,26,30)(22,27,25,33)(24,31,32,35)(28,29,34,36);(6,12)(8,9)(11,18)(13,14)(15,17)(19,20)(21,24,26,32)(22,28,25,34)(23,29,30,36)(27,35,33,31);(1,3,4,2)(5,8,10,9)(6,13,14,12)(7,15,16,17)(11,18,20,19)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=i,d^2=eh,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=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169716);