#7589330

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 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)(17,18,22,27)(19,28,32,24)(20,25,35,45)(21,36,23,26)(29,43,33,39)(30,47,48,40)(31,41,38,42)(34,46,37,44);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,28)(20,33,35,29)(21,31,23,38)(25,43,45,39)(26,41,36,42)(30,37,48,34)(40,44,47,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,39)(26,46)(27,45)(28,43)(30,31)(32,33)(36,44)(38,48)(40,42)(41,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,40)(26,44)(27,45)(28,47)(29,31)(32,48)(33,38)(36,46)(39,42)(41,43);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,36)(19,31,32,38)(20,34,35,37)(24,41,28,42)(25,44,45,46)(29,48,33,30)(39,40,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7496526);