#392472

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1498 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)(7,8)(9,10,15,19,14,17,13,20)(11,21,25,30,24,32,23,16)(12,26,27,18,28,34,29,33)(22,36,37,35,38,31,39,40);(9,11,13,23,14,24,15,25)(10,16,19,21,17,30,20,32)(12,22,27,37,28,38,29,39)(18,31,26,35,33,36,34,40);(1,2,3,4)(10,17)(16,30)(18,33)(19,20)(21,32)(26,34)(31,36)(35,40);(6,8)(9,12)(10,18)(11,22)(13,27)(14,28)(15,29)(16,31)(17,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(30,36)(32,40)

Polycyclic presentation:

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

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, 299668);