#8728398

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

4263 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,43)(36,42)(37,40)(38,44)(39,41)(45,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,37)(34,40,41,43)(36,47,44,45)(38,46,42,48);(3,10)(6,7)(11,16)(14,15)(20,28)(24,29)(26,32)(30,31)(34,41)(36,44)(40,43)(45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31)(33,36)(34,42)(35,45)(37,47)(38,41)(39,44)(40,46)(43,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30)(33,37,39,35)(34,43,41,40)(36,47,44,45)(38,46,42,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)(33,38,39,42)(34,44,41,36)(35,46,37,48)(40,45,43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635594);