#393759

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 300955);