#7581931

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,38)(29,39)(31,47)(32,44)(33,40)(34,41)(35,42)(36,43)(37,45)(46,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,20)(18,24,26,30)(21,33,34,46)(23,32,36,29)(25,40,41,48)(27,44,43,39)(28,37,31,35)(38,45,47,42);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,19)(18,24,26,30)(21,31,34,28)(23,32,36,29)(25,38,41,47)(27,39,43,44)(33,35,46,37)(40,45,48,42);(10,14)(13,15)(17,21,22,34)(18,25,26,41)(19,28,20,31)(23,37,36,35)(24,38,30,47)(27,45,43,42)(29,46,32,33)(39,48,44,40);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,20)(24,30)(28,31)(29,32)(33,46)(38,47)(39,44)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7489127);