#6357308

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357300
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1844 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:

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

Polycyclic presentation:

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

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, 6264504);