#7590026

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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)(19,29,30,24)(20,25,34,40)(21,26,23,28)(31,43,44,37)(32,41,35,38)(33,39,36,42)(45,48,46,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,30)(20,33)(21,23)(24,29)(25,39)(26,28)(31,45)(34,36)(37,47)(40,42)(43,48)(44,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,30)(18,24,27,29)(20,31,34,44)(21,35,23,32)(25,37,40,43)(26,41,28,38)(33,46,36,45)(39,48,42,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,31)(21,33)(22,34)(23,36)(24,37)(26,39)(27,40)(28,42)(29,43)(30,44)(32,45)(35,46)(38,47)(41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,32,30,35)(20,33,34,36)(24,38,29,41)(25,39,40,42)(31,45,44,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fg,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7497222);