#7623445

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	#60695	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1237 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)(18,23)(19,27)(21,33)(22,32)(24,38)(25,41)(28,39)(29,46)(30,47)(31,45)(34,35)(36,43)(40,42)(44,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18)(19,28)(20,23)(21,25)(22,26)(24,29)(27,38)(30,44)(31,36)(32,37)(33,43)(34,48)(35,40)(39,46)(41,45)(42,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,20,29)(18,24,23,28)(21,34,31,47)(22,35,32,30)(25,42,36,48)(26,44,37,40)(27,45,46,33)(38,43,39,41);(2,8)(6,9)(7,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,39)(28,29)(30,40)(31,36)(32,37)(33,43)(34,42)(35,44)(38,46)(41,45)(47,48);(18,23)(24,28)(25,36)(26,37)(38,39)(40,44)(41,43)(42,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,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,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, 7530641);