#8587168

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

4024 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8494364);