#6516518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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

1382 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:

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

Polycyclic presentation:

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

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, 6423714);