#8206119

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

4056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8113315);