#6352123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259319);