#7207043

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(10,14)(11,12)(17,18,21,23)(19,27,29,40)(20,25,31,37)(22,34,33,26)(24,35,36,30)(28,41,42,48)(32,45,44,39)(38,46,47,43);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(18,23)(24,36)(25,37)(26,34)(27,40)(38,47)(39,45)(41,48);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(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)(9,11,13,12)(10,16,14,15)(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)(9,12,13,11)(10,16,14,15)(18,23)(19,29)(20,31)(26,34)(30,35)(32,44)(38,47)(41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7114239);