#7606648

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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)(11,15)(12,13)(14,16)(17,18)(19,26)(20,25)(21,36)(22,38)(23,28)(24,32)(27,29)(30,43)(31,44)(33,40)(34,39)(35,47)(37,48)(41,46)(42,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,19,29,21,23,31,22,32)(18,24,38,44,28,36,27,26)(20,30,45,37,33,46,35,34)(25,39,47,41,40,48,42,43);(1,4,5,2)(3,7,8,6)(10,13)(14,15)(17,20,23,33)(18,25,28,40)(19,30,31,46)(21,37,32,34)(22,35,29,45)(24,39,36,48)(26,43,44,41)(27,42,38,47);(3,8)(6,7)(9,11,12,16)(10,14,13,15)(19,31)(20,33)(21,32)(24,36)(25,40)(26,44)(35,45)(42,47);(17,21,29,31,23,32,22,19)(18,26,38,24,28,44,27,36)(20,34,45,30,33,37,35,46)(25,41,47,48,40,43,42,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=gh,d^2=g,e^2=fi,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,e^c=ei,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, 7513844);