#7160720

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7067916);