#403044

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=e,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310240);