#6453382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(11,14)(12,15)(13,16)(17,18,23,27)(19,24,31,40)(20,25,34,44)(21,35,22,26)(28,37,46,47)(29,38,30,39)(32,41,45,48)(33,43,36,42);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19,23,31)(18,24,27,40)(20,32,34,45)(21,30,22,29)(25,41,44,48)(26,38,35,39)(28,36,46,33)(37,42,47,43);(9,11,13,12)(10,15,16,14)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,42)(27,44)(29,32)(30,45)(31,46)(35,43)(38,41)(39,48)(40,47);(9,12,13,11)(10,15,16,14)(19,29)(21,22)(24,38)(26,35)(28,32)(30,31)(33,36)(37,41)(39,40)(42,43)(45,46)(47,48);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,35)(19,30,31,29)(20,33,34,36)(24,38,40,39)(25,42,44,43)(28,45,46,32)(37,41,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360578);