#6352402

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	#62878	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^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, 6259598);