#6372673

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279869);