#393399

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,23)(16,28)(17,31)(18,20)(19,26)(21,30)(22,24)(25,39)(27,35)(29,34)(32,37)(33,44)(36,41)(38,43)(40,42);(1,3,4,2)(5,7,9,8)(6,10,11,12)(13,15,24,37,18,27,17,26)(14,19,31,35,20,32,22,23)(16,25,38,44,30,41,29,40)(21,33,43,39,28,42,34,36);(5,8,9,7)(6,10,11,12)(14,20)(19,32)(21,28)(22,31)(23,35)(33,42)(34,43)(36,39);(13,16,17,29,18,30,24,38)(14,21,22,34,20,28,31,43)(15,25,26,40,27,41,37,44)(19,33,23,36,32,42,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300595);