#7513548

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,38)(29,37)(30,39)(31,41)(32,40)(33,43)(34,44)(35,42)(45,47)(46,48);(1,3)(2,4)(5,7)(6,8)(10,13)(11,15)(17,19,22,29)(18,23,25,37)(20,30,32,46)(21,33,35,28)(24,39,40,48)(26,43,42,38)(27,34,45,31)(36,44,47,41);(10,13)(14,16)(17,20,22,32)(18,24,25,40)(19,27,29,45)(21,34,35,31)(23,36,37,47)(26,44,42,41)(28,46,33,30)(38,48,43,39);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16);(1,5,6,4)(2,3,7,8)(18,25)(23,37)(24,40)(26,42)(36,47)(38,43)(39,48)(41,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7420744);