#7589321

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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,21,25)(19,27,29,23)(20,31,33,24)(22,26,34,39)(28,36,44,41)(30,42,45,37)(32,40,35,38)(43,47,46,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(20,32)(22,34)(24,38)(26,39)(28,43)(30,45)(31,40)(33,35)(36,47)(37,42)(41,48)(44,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,41)(32,43)(33,44)(34,45)(35,46)(38,47)(39,42)(40,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,21,33)(18,24,25,31)(19,28,29,44)(22,35,34,32)(23,36,27,41)(26,40,39,38)(30,46,45,43)(37,48,42,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,29)(23,27)(28,44)(30,45)(36,41)(37,42)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fg,e^2=fgi,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, 7496517);