#6515008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,24,22,37)(20,30,33,38)(21,26,36,44)(25,31,41,28)(29,39,34,42)(32,47,35,48)(40,46,43,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,34,36,29)(25,30,41,38)(26,42,44,39)(32,45,35,46)(40,48,43,47);(10,13)(11,15)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,35,36,32)(24,38,37,30)(26,43,44,40)(29,46,34,45)(39,47,42,48);(9,11,12,15)(10,14,13,16)(17,21,23,36)(18,26,27,44)(19,29,22,34)(20,32,33,35)(24,39,37,42)(25,40,41,43)(28,45,31,46)(30,47,38,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(21,36)(26,44)(29,34)(32,35)(39,42)(40,43)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,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, 6422204);