#8505502

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

4110 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412698);