#7545322

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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)(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,25)(21,34)(22,26)(23,27)(28,36)(29,37)(30,38)(31,39)(32,40)(33,42)(35,41)(43,46)(44,48)(45,47);(2,6)(7,8)(9,11,13,10)(12,14,16,15)(17,19,23,31)(18,24,27,39)(20,28,32,45)(21,30,22,29)(25,36,40,47)(26,37,34,38)(33,44,35,43)(41,46,42,48);(1,3,5,4)(2,7,6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,28)(21,35)(22,33)(23,32)(24,36)(26,42)(27,40)(29,44)(30,43)(31,45)(34,41)(37,48)(38,46)(39,47);(1,4,5,3)(2,8,6,7)(10,11)(14,15)(19,29)(20,32)(21,22)(24,37)(25,40)(26,34)(28,43)(30,31)(36,46)(38,39)(44,45)(47,48);(2,6)(7,8)(17,21,23,22)(18,26,27,34)(19,30,31,29)(20,33,32,35)(24,37,39,38)(25,41,40,42)(28,44,45,43)(36,46,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=f,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, 7452518);