#7595486

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,31,20,25)(21,36,39,46)(22,30,24,28)(26,34,42,33)(27,40,45,38)(32,43,35,44)(37,41,48,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,23)(21,37)(22,33)(24,34)(26,28)(27,43)(29,31)(30,42)(32,40)(35,38)(36,47)(39,48)(41,46)(44,45);(10,14)(11,16)(17,20,24,34,23,19,22,33)(18,26,30,31,29,42,28,25)(21,32,40,37,39,35,38,48)(27,44,46,41,45,43,36,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,40,23,39,22,38)(18,27,30,46,29,45,28,36)(19,32,33,37,20,35,34,48)(25,41,26,43,31,47,42,44);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,33,20,34)(21,38,39,40)(25,26,31,42)(27,36,45,46)(32,37,35,48)(41,43,47,44)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei

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, 7502682);