#8728990

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

4009 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34,38,41)(35,40,44,39)(36,45,46,37)(42,43,48,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(19,26)(22,25)(27,28)(31,32)(33,35,38,44)(34,39,41,40)(36,42,46,48)(37,43,45,47);(3,10)(6,7)(11,16)(14,15)(17,19,21,26)(18,22,24,25)(20,27,30,28)(23,31,29,32)(33,35,38,44)(34,39,41,40)(36,42,46,48)(37,43,45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,35)(34,40)(36,42)(37,47)(38,44)(39,41)(43,45)(46,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,36,38,46)(34,37,41,45)(35,42,44,48)(39,43,40,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,20,21,30)(18,23,24,29)(19,27,26,28)(22,31,25,32)(33,37)(34,36)(35,43)(38,45)(39,42)(40,48)(41,46)(44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=gh,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, 8636186);