#8471066

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8378262);