#391904

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299100);