#7623425

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623308
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi

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, 7530621);