#7582431

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7489627);