#8184880

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092076);