#7513729

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,35)(21,26)(22,27)(23,28)(24,29)(30,46)(31,40)(32,41)(33,42)(34,43)(36,48)(37,45)(38,44)(39,47);(1,3,6,2)(4,7,5,8)(17,19,23,33)(18,25,28,42)(20,36,21,31)(22,34,24,32)(26,40,35,48)(27,43,29,41)(30,38,47,37)(39,45,46,44);(10,13)(14,16)(17,20,23,21)(18,26,28,35)(19,30,33,47)(22,37,24,38)(25,39,42,46)(27,44,29,45)(31,32,36,34)(40,43,48,41);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,21,23,20)(18,26,28,35)(19,31,33,36)(22,37,24,38)(25,40,42,48)(27,45,29,44)(30,32,47,34)(39,43,46,41);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,28,29)(19,32,33,34)(20,37,21,38)(25,41,42,43)(26,44,35,45)(30,36,47,31)(39,40,46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7420925);