#262512

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	#59822	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(5,6,9,13)(7,15,17,11)(8,14,10,12)(16,20,18,19)(21,22,26,25)(23,27,30,33)(24,28,32,34)(29,35,36,31);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,16,10,18)(12,20,14,19)(21,23,26,30)(22,27,25,33)(24,31,32,35)(28,29,34,36);(6,12)(8,10)(11,19)(13,14)(15,20)(16,18)(21,24,26,32)(22,28,25,34)(23,29,30,36)(27,35,33,31);(1,3,4,2)(5,8,9,10)(6,12,13,14)(7,16,17,18)(11,20,15,19)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=i,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=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, 169708);