#6516443

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,28,31,39)(20,25,32,41)(21,26,22,36)(24,37,40,30)(29,45,33,48)(34,43,35,44)(38,46,42,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(20,32)(25,41)(29,33)(34,35)(38,42)(43,44)(45,48)(46,47);(10,13)(11,15)(17,19,23,31)(18,24,27,40)(20,33,32,29)(21,30,22,37)(25,42,41,38)(26,39,36,28)(34,47,35,46)(43,48,44,45);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,23,32)(18,25,27,41)(19,29,31,33)(21,35,22,34)(24,38,40,42)(26,44,36,43)(28,45,39,48)(30,47,37,46);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,32,35)(24,39,40,28)(25,43,41,44)(29,46,33,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6423639);