#7207179

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

Polycyclic presentation:

2^9|a^2=i,b^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=ehi,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, 7114375);