#7620468

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fghi,e^2=gi,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, 7527664);