#6352076

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2847 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:

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

Polycyclic presentation:

2^9|b^2=h,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, 6259272);