#7596004

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(12,13)(14,16)(17,18,23,28)(19,30,32,25)(20,34,38,45)(21,40,33,42)(22,29,24,27)(26,37,46,39)(31,47,48,43)(35,44,36,41);(2,6)(7,8)(10,13)(11,16)(17,19)(18,25)(20,35)(21,22)(23,32)(24,33)(26,43)(27,40)(28,30)(29,42)(31,39)(34,41)(36,38)(37,48)(44,45)(46,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,37,23,38,24,39)(18,26,27,45,28,46,29,34)(19,31,33,36,32,48,21,35)(25,41,42,47,30,44,40,43);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,24,32,23,33,22,19)(18,25,29,40,28,30,27,42)(20,36,39,31,38,35,37,48)(26,44,34,47,46,41,45,43);(2,6)(7,8)(10,13)(14,15)(17,22,23,24)(18,27,28,29)(19,21,32,33)(20,37,38,39)(25,40,30,42)(26,45,46,34)(31,35,48,36)(41,43,44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=fgi,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^d=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, 7503200);