#8184787

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
prime radical	#58307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(34,39)(40,41)(42,46)(47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,38,39)(35,40,36,41)(37,42,45,46)(43,47,44,48);(2,7)(5,8)(9,12)(13,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,35,38,36)(34,40,39,41)(37,44,45,43)(42,48,46,47);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,45)(34,42,39,46)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 8091983);