#7051943

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6959139);