#7147306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7054502);