#7617326

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 ◁ #2399 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=g,d^2=hi,e^2=fi,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, 7524522);