#6825859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6733055);