#7531698

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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)(11,15)(12,14)(13,16)(17,18)(19,26)(20,25)(21,34)(22,39)(23,29)(24,28)(27,37)(30,40)(31,41)(32,43)(33,42)(35,44)(36,45)(38,46)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,23)(21,35)(22,31)(24,33)(26,29)(27,44)(28,41)(30,38)(32,36)(34,48)(37,47)(39,42)(40,45)(43,46);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,23,19)(18,26,29,25)(21,32,37,30)(22,31,24,33)(27,43,34,40)(28,41,39,42)(35,38,47,36)(44,45,48,46);(10,14)(13,15)(17,21,23,37)(18,27,29,34)(19,30,20,32)(22,38,24,36)(25,40,26,43)(28,46,39,45)(31,47,33,35)(41,44,42,48);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,31,20,33)(21,36,37,38)(25,41,26,42)(27,45,34,46)(30,35,32,47)(40,48,43,44)

Polycyclic presentation:

2^9|c^2=fgi,d^2=i,e^2=i,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,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, 7438894);