#6472755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,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, 6379951);