#6471875

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,34,21,26)(23,38,37,28)(25,36,42,35)(30,40,31,44)(33,41,48,45)(39,46,43,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,27,29)(20,30,21,31)(23,33,37,48)(25,39,42,43)(26,40,34,44)(28,41,38,45)(35,46,36,47);(11,15)(13,16)(17,20,22,21)(18,25,27,42)(19,30,32,31)(23,36,37,35)(24,39,29,43)(26,38,34,28)(33,47,48,46)(40,45,44,41);(11,15)(13,16)(17,21,22,20)(18,26,27,34)(19,31,32,30)(23,36,37,35)(24,40,29,44)(25,38,42,28)(33,47,48,46)(39,45,43,41);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(20,21)(25,42)(26,34)(30,31)(35,36)(39,43)(40,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,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^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, 6379071);