#6753280

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(3,7)(4,6)(5,8)(17,18,22,26)(19,23,28,38)(20,30,32,42)(21,34,36,25)(24,35,41,33)(27,43,31,48)(29,46,45,39)(37,47,40,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,28)(18,23,26,38)(20,31,32,27)(21,29,36,45)(24,40,41,37)(25,39,34,46)(30,48,42,43)(33,47,35,44);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,35)(22,32)(23,37)(25,42)(26,41)(28,31)(29,47)(30,34)(33,36)(38,40)(39,48)(43,46)(44,45);(2,6)(5,7)(19,28)(23,38)(27,31)(29,45)(37,40)(39,46)(43,48)(44,47);(10,14)(13,15)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6660476);