#6448527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18,23,28)(19,24,32,40)(20,25,35,43)(21,26,22,27)(29,37,31,39)(30,38,46,48)(33,41,45,47)(34,42,36,44);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,40)(20,33,35,45)(21,31,22,29)(25,41,43,47)(26,39,27,37)(30,34,46,36)(38,42,48,44);(9,12,13,11)(10,14,15,16)(19,29)(21,22)(24,37)(26,27)(30,45)(31,32)(33,46)(34,36)(38,47)(39,40)(41,48)(42,44);(17,20,23,35)(18,25,28,43)(19,30,32,46)(21,34,22,36)(24,38,40,48)(26,42,27,44)(29,45,31,33)(37,47,39,41);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,28,27)(19,31,32,29)(20,34,35,36)(24,39,40,37)(25,42,43,44)(30,33,46,45)(38,41,48,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355723);