#7204162

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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)(19,27)(20,25)(21,23)(22,34)(24,30)(26,33)(28,41)(29,40)(31,37)(32,45)(35,36)(38,43)(39,44)(42,48)(46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(18,23)(20,31)(24,36)(26,34)(27,40)(28,42)(32,44)(43,46);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,28)(21,29)(22,35)(23,36)(25,38)(26,40)(27,34)(30,33)(31,42)(32,46)(37,47)(39,48)(41,45)(43,44);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,20,21,31)(18,25,23,37)(19,28,29,42)(22,32,33,44)(24,38,36,47)(26,39,34,45)(27,41,40,48)(30,43,35,46);(3,7)(5,8)(10,15)(14,16)(18,23)(19,29)(20,31)(26,34)(30,35)(32,44)(38,47)(41,48)

Polycyclic presentation:

2^9|a^2=fi,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, 7111358);