#8551363

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

4178 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8458559);