#8621215

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

4022 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528411);