#6475500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1705 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,27,31,40)(20,25,32,38)(21,34,28,26)(23,35,36,30)(29,41,43,48)(33,45,42,39)(37,46,47,44);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,28,35,22,31,21,30)(18,23,34,40,24,36,26,27)(20,29,42,46,32,43,33,44)(25,37,45,48,38,47,39,41);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,24)(20,32)(23,36)(26,34)(27,40)(29,43)(33,42)(44,46);(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,38)(26,39)(27,41)(28,42)(30,44)(31,43)(34,45)(35,46)(36,47)(40,48);(10,14)(13,16)(20,32)(25,38)(29,43)(33,42)(37,47)(39,45)(41,48)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6382696);