#7162168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(11,15)(14,16)(17,18,22,26)(19,27,30,43)(20,24,32,40)(21,34,35,25)(23,36,38,29)(28,42,31,47)(33,48,44,41)(37,45,39,46);(2,5)(3,7)(10,13)(11,15)(17,19,21,29,22,30,35,36)(18,23,25,27,26,38,34,43)(20,31,33,45,32,28,44,46)(24,39,41,47,40,37,48,42);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,42)(29,46)(30,31)(34,48)(35,44)(36,45)(38,39)(43,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,32)(21,35)(23,27)(24,40)(25,34)(28,45)(30,36)(31,46)(37,47)(38,43)(39,42);(9,11,12,15)(10,14,13,16)(19,30)(20,32)(23,38)(24,40)(27,43)(29,36)(33,44)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7069364);