#7608144

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1257 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gh,d^2=gi,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

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, 7515340);