#6545494

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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)(17,18,23,35,22,27,21,26)(19,24,36,46,29,38,30,39)(20,31,32,25,34,40,33,41)(28,42,44,37,43,48,45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,27,38)(20,28,34,43)(21,30,23,36)(25,37,41,47)(26,39,35,46)(31,42,40,48)(32,44,33,45);(10,13)(14,16)(17,20)(18,25)(19,28)(21,32)(22,34)(23,33)(24,37)(26,40)(27,41)(29,43)(30,44)(31,35)(36,45)(38,47)(39,48)(42,46);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,29)(24,38)(28,43)(30,36)(37,47)(39,46)(42,48)(44,45);(11,15)(14,16)(17,21,22,23)(18,26,27,35)(19,30,29,36)(20,32,34,33)(24,39,38,46)(25,40,41,31)(28,44,43,45)(37,48,47,42)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452690);