#8271076

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

4067 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=hi,f^2=gi,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, 8178272);