#6516277

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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)(3,7)(4,6)(5,8)(17,18,22,25)(19,23,28,38)(20,30,32,42)(21,34,36,26)(24,35,41,33)(27,43,31,48)(29,46,45,39)(37,47,40,44);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,25,38)(20,31,32,27)(21,29,36,45)(24,40,41,37)(26,39,34,46)(30,48,42,43)(33,47,35,44);(10,13)(11,15)(17,20)(18,24)(19,27)(21,35)(22,32)(23,37)(25,41)(26,42)(28,31)(29,47)(30,34)(33,36)(38,40)(39,48)(43,46)(44,45);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(18,25)(19,28)(24,41)(26,34)(27,31)(29,45)(30,42)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

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, 6423473);