#6516723

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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, 6423919);