#7409997

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,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, 7317193);