#6472000

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	#24455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 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)(17,18,22,25)(19,23,28,39)(20,30,32,42)(21,34,36,26)(24,35,41,33)(27,43,31,48)(29,46,45,38)(37,47,40,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,28)(18,23,25,39)(20,31,32,27)(21,29,36,45)(24,40,41,37)(26,38,34,46)(30,48,42,43)(33,47,35,44);(11,15)(13,16)(17,20,22,32)(18,24,25,41)(19,27,28,31)(21,35,36,33)(23,37,39,40)(26,42,34,30)(29,47,45,44)(38,48,46,43);(11,15)(13,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,37)(25,39)(26,38)(30,43)(31,32)(33,44)(34,46)(35,47)(36,45)(40,41)(42,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,25)(19,28)(20,32)(26,34)(29,45)(33,35)(37,40)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6379196);