#7614713

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,44)(31,41)(33,46)(34,40)(35,42)(36,48)(37,43)(38,45)(39,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,30,31,47)(21,36,37,35)(23,34,38,33)(25,39,41,44)(26,42,43,48)(28,46,45,40);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,30)(21,34)(22,31)(23,35)(24,39)(26,40)(27,41)(28,42)(29,44)(32,47)(33,37)(36,38)(43,46)(45,48);(10,14)(13,15)(17,21,23,30,22,37,38,47)(18,26,28,44,27,43,45,39)(19,31,33,36,32,20,34,35)(24,25,40,42,29,41,46,48);(2,5)(6,8)(10,14)(13,15)(18,27)(20,31)(21,37)(24,29)(28,45)(30,47)(35,36)(40,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521909);