#6489170

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25727	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396366);