#7204154

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#53053	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,30)(21,26)(22,35)(23,27)(25,28)(29,38)(31,43)(32,37)(33,41)(34,45)(36,40)(39,42)(44,47)(46,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,33,34)(25,37,40,30)(26,38,41,45)(31,42,44,46)(39,47,48,43);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,33,44)(24,37,35,30)(26,39,41,48)(29,42,34,46)(38,47,45,43);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,21)(18,26)(19,29)(20,31)(22,34)(23,33)(24,38)(25,39)(27,41)(28,42)(30,43)(32,44)(35,45)(36,46)(37,47)(40,48);(3,7)(5,8)(10,15)(14,16)(20,32)(21,33)(25,40)(26,41)(28,36)(29,34)(30,37)(38,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7111350);