#6489208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23941	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1854 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,23,28,24,29,22,27)(19,30,33,40,34,43,38,25)(20,35,36,42,21,26,37,41)(31,45,47,39,32,44,46,48);(11,12)(14,15)(17,19)(18,25)(20,31)(21,32)(22,33)(23,38)(24,34)(26,39)(27,30)(28,43)(29,40)(35,48)(36,46)(37,47)(41,45)(42,44);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,24,21)(18,26,29,35)(19,31,34,32)(22,37,23,36)(25,39,40,48)(27,42,28,41)(30,44,43,45)(33,47,38,46);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,26,29,35)(19,32,34,31)(22,36,23,37)(25,39,40,48)(27,42,28,41)(30,44,43,45)(33,46,38,47);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,22,24,23)(18,27,29,28)(19,33,34,38)(20,36,21,37)(25,30,40,43)(26,41,35,42)(31,46,32,47)(39,45,48,44)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6396404);