#8184800

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

4130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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, 8091996);