#7609943

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,31,24)(20,33,35,25)(21,37,38,26)(23,28,39,47)(30,41,34,43)(32,44,40,42)(36,46,48,45);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,29)(20,34,35,30)(21,36,38,48)(23,40,39,32)(25,43,33,41)(26,46,37,45)(28,44,47,42);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,22,35)(18,25,27,33)(19,30,31,34)(21,32,38,40)(23,36,39,48)(24,41,29,43)(26,44,37,42)(28,45,47,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,23,30,22,38,39,34)(18,26,28,43,27,37,47,41)(19,20,32,36,31,35,40,48)(24,33,42,46,29,25,44,45);(2,6)(7,8)(18,27)(20,35)(21,38)(24,29)(28,47)(30,34)(36,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7517139);