#7623429

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
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	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1228 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)(17,18,20,25)(19,28,29,23)(21,24,32,40)(22,35,33,27)(26,42,41,34)(30,36,45,39)(31,38,46,44)(37,48,43,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,29)(18,23,25,28)(21,30,32,45)(22,31,33,46)(24,39,40,36)(26,37,41,43)(27,38,35,44)(34,48,42,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,29)(21,34)(22,33)(23,36)(25,40)(26,27)(28,39)(30,47)(32,42)(35,41)(37,44)(38,43)(45,48);(2,8)(6,9)(7,16)(14,15)(18,25)(23,28)(24,40)(26,41)(27,35)(36,39)(37,43)(38,44);(19,29)(23,28)(30,45)(31,46)(36,39)(37,43)(38,44)(47,48)

Polycyclic presentation:

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

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, 7530625);