#8271074

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

4090 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 8178270);