#7612715

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1369 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:

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

Polycyclic presentation:

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

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, 7519911);