#8425524

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

4170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332720);