#7582271

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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,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, 7489467);