#7544236

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,42)(32,46)(33,39)(34,40)(35,48)(36,43)(37,44)(38,45)(41,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,20)(18,25,28,30)(21,35,37,47)(22,32,24,34)(26,41,44,48)(27,40,29,46)(31,36,33,38)(39,43,42,45);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,23,19)(18,25,28,30)(21,31,37,33)(22,32,24,34)(26,42,44,39)(27,46,29,40)(35,36,47,38)(41,45,48,43);(10,14)(13,15)(17,21,23,37)(18,26,28,44)(19,31,20,33)(22,38,24,36)(25,39,30,42)(27,45,29,43)(32,47,34,35)(40,48,46,41);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,32,20,34)(21,36,37,38)(25,40,30,46)(26,43,44,45)(31,35,33,47)(39,41,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7451432);