#7531689

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 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)(17,18,22,23)(19,24,29,37)(20,30,33,25)(21,34,36,26)(27,40,31,38)(28,35,43,39)(32,41,47,44)(42,46,45,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,23,37)(20,31,33,27)(21,35,36,39)(25,40,30,38)(26,28,34,43)(32,46,47,48)(41,45,44,42);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,27)(21,32)(22,33)(23,30)(24,38)(26,41)(28,42)(29,31)(34,44)(35,48)(36,47)(37,40)(39,46)(43,45);(10,14)(13,15)(17,21,22,36)(18,26,23,34)(19,28,29,43)(20,32,33,47)(24,39,37,35)(25,41,30,44)(27,42,31,45)(38,46,40,48);(2,5)(6,8)(10,14)(13,15)(20,33)(25,30)(27,31)(32,47)(38,40)(41,44)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^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,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, 7438885);