#8505614

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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gi,e^2=h,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, 8412810);