#7089809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,24,29,39)(20,25,33,41)(21,23,34,26)(28,38,32,42)(30,36,45,40)(31,43,35,37)(44,48,46,47);(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(21,34)(24,36)(25,41)(26,27)(28,44)(30,45)(32,46)(33,35)(38,42)(39,40);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,29)(23,36)(25,42)(26,40)(27,39)(28,33)(31,46)(34,45)(35,44)(37,48)(38,41)(43,47);(11,16)(13,15)(17,20,22,33)(18,25,27,41)(19,28,29,32)(21,35,34,31)(23,37,26,43)(24,38,39,42)(30,46,45,44)(36,47,40,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,29)(24,39)(28,32)(30,45)(36,40)(38,42)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997005);