#7532807

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

1505 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)(19,23,29,40)(20,31,33,24)(22,26,35,44)(27,42,32,38)(28,47,48,39)(30,41,36,45)(34,46,37,43);(2,6)(7,8)(11,16)(13,15)(17,19,21,29)(18,23,25,40)(20,32,33,27)(22,36,35,30)(24,42,31,38)(26,45,44,41)(28,37,48,34)(39,46,47,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,33)(22,34)(23,38)(25,31)(26,43)(28,30)(29,32)(35,37)(36,48)(39,41)(40,42)(44,46)(45,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,28)(21,33)(22,37)(23,39)(25,31)(26,46)(27,30)(29,48)(32,36)(34,35)(38,41)(40,47)(42,45)(43,44);(2,6)(7,8)(20,33)(24,31)(27,32)(28,48)(34,37)(38,42)(39,47)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,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, 7440003);