#6352464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352341
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	#18304	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,23,22)(19,24,30,29)(20,25,33,32)(21,26,34,35)(27,39,41,36)(28,37,42,43)(31,38,44,45)(40,47,48,46);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,36)(26,37)(31,40)(32,39)(33,41)(34,42)(35,43)(38,46)(44,48)(45,47);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,20)(18,25)(19,27)(21,31)(22,32)(23,33)(24,36)(26,38)(28,40)(29,39)(30,41)(34,44)(35,45)(37,46)(42,48)(43,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,34)(18,26,22,35)(19,28,30,42)(20,31,33,44)(24,37,29,43)(25,38,32,45)(27,40,41,48)(36,46,39,47);(17,22,23,18)(19,29,30,24)(20,32,33,25)(21,26,34,35)(27,39,41,36)(28,37,42,43)(31,38,44,45)(40,46,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259660);