#333368

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=i,c^2=hi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240564);