#7543386

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1332 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)(11,14)(12,16)(13,15)(17,18,23,28)(19,24,32,41)(20,25,35,44)(21,26,22,27)(29,38,33,42)(30,39,47,48)(31,40,37,46)(34,43,36,45);(2,6)(7,8)(9,11,14,10)(12,16,13,15)(17,19,23,32)(18,24,28,41)(20,33,35,29)(21,31,22,37)(25,42,44,38)(26,40,27,46)(30,36,47,34)(39,45,48,43);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,45)(27,43)(28,44)(30,31)(32,33)(37,47)(39,40)(41,42)(46,48);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,39)(26,43)(27,45)(28,44)(29,31)(32,47)(33,37)(38,40)(41,48)(42,46);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,11,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,35,36)(24,40,41,46)(25,43,44,45)(29,47,33,30)(38,48,42,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fg,d^2=f,e^2=fi,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, 7450582);