#7532931

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,30,38)(20,24,31,41)(21,26,35,42)(27,36,45,48)(28,37,46,47)(29,39,33,43)(32,40,34,44);(2,6)(7,8)(11,16)(13,15)(17,19,22,30)(18,23,25,38)(20,27,31,45)(21,33,35,29)(24,36,41,48)(26,43,42,39)(28,34,46,32)(37,44,47,40);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,36)(25,41)(26,40)(28,33)(29,46)(30,45)(34,35)(37,43)(38,48)(39,47)(42,44);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,28)(21,34)(22,31)(23,37)(25,41)(26,44)(27,33)(29,45)(30,46)(32,35)(36,43)(38,47)(39,48)(40,42);(2,6)(7,8)(18,25)(20,31)(23,38)(26,42)(27,45)(28,46)(32,34)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,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, 7440127);