#6489173

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,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,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, 6396369);