#7590006

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(20,33,35,25)(21,26,23,28)(31,38,44,43)(32,41,36,39)(34,42,37,40)(45,47,46,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,30)(20,34)(21,23)(24,29)(25,40)(26,28)(31,45)(33,42)(35,37)(38,47)(43,48)(44,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,22,30)(18,24,27,29)(20,31,35,44)(21,36,23,32)(25,38,33,43)(26,41,28,39)(34,46,37,45)(40,48,42,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,31)(21,34)(22,35)(23,37)(24,38)(26,40)(27,33)(28,42)(29,43)(30,44)(32,45)(36,46)(39,47)(41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,32,30,36)(20,34,35,37)(24,39,29,41)(25,40,33,42)(31,45,44,46)(38,47,43,48)

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=dg,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497202);