#7548468

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

1402 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,27)(19,29,33,24)(20,25,34,40)(21,26,23,28)(30,43,46,37)(31,39,32,38)(35,41,36,42)(44,48,45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,33)(18,24,27,29)(20,30,34,46)(21,32,23,31)(25,37,40,43)(26,39,28,38)(35,45,36,44)(41,48,42,47);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,42)(27,40)(28,41)(29,43)(31,45)(32,44)(33,46)(38,48)(39,47);(9,11,12,16)(10,14,13,15)(19,31)(20,34)(21,23)(24,38)(25,40)(26,28)(29,39)(30,44)(32,33)(37,47)(43,48)(45,46);(2,7)(6,8)(17,21,22,23)(18,26,27,28)(19,32,33,31)(20,35,34,36)(24,39,29,38)(25,41,40,42)(30,45,46,44)(37,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=g,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, 7455664);