#6468353

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(9,10,13,16)(11,14,12,15)(17,18)(19,29)(20,25)(21,26)(22,37)(23,28)(24,32)(27,34)(30,43)(31,38)(33,44)(35,41)(36,42)(39,46)(40,45)(47,48);(11,12)(14,15)(17,19,22,32,23,33,34,45)(18,24,27,29,28,40,37,44)(20,30,21,31,35,46,36,47)(25,38,42,43,41,48,26,39);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,30)(21,22)(23,35)(24,38)(26,37)(27,42)(28,41)(29,43)(31,32)(33,46)(34,36)(39,44)(40,48)(45,47);(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,31)(20,34)(22,35)(23,36)(24,39)(25,27)(28,42)(29,38)(30,45)(32,46)(33,47)(37,41)(40,43)(44,48);(1,3,4,7)(2,6,5,8)(20,35)(21,36)(25,41)(26,42)(30,46)(31,47)(38,48)(39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375549);