#7205960

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(17,18,22,27)(19,29,32,24)(20,26,35,42)(21,28,23,25)(30,43,45,38)(31,37,33,39)(34,40,36,41)(44,48,46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,32)(18,24,27,29)(20,30,35,45)(21,31,23,33)(25,37,28,39)(26,38,42,43)(34,44,36,46)(40,47,41,48);(3,7)(5,8)(10,14)(15,16)(18,25)(21,23)(24,37)(26,40)(27,28)(29,39)(31,33)(34,36)(38,47)(41,42)(43,48)(44,46);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,38)(25,40)(27,42)(28,41)(29,43)(31,46)(32,45)(33,44)(37,47)(39,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,25,27,28)(19,31,32,33)(20,34,35,36)(24,37,29,39)(26,41,42,40)(30,44,45,46)(38,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,e^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=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, 7113156);