#8730893

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

4136 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)(3,5)(4,11)(6,7)(8,13)(9,15)(10,12)(14,16)(17,18,22,26)(19,23,27,24)(20,21,25,31)(28,29,32,30)(33,34)(35,41)(36,38)(37,40)(39,42)(43,47)(44,45)(46,48);(2,7)(3,9)(8,11)(10,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31)(33,35,39,37)(34,40,42,41)(36,43,46,45)(38,44,48,47);(2,7)(3,9)(8,11)(10,16)(19,27)(23,24)(28,32)(29,30)(33,35,39,37)(34,41,42,40)(36,45,46,43)(38,44,48,47);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,19)(18,24)(20,28)(21,30)(22,27)(23,26)(25,32)(29,31)(33,36,39,46)(34,38,42,48)(35,43,37,45)(40,44,41,47);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,20)(18,21)(19,28)(22,25)(23,29)(24,30)(26,31)(27,32)(33,37,39,35)(34,40,42,41)(36,45,46,43)(38,44,48,47);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)(33,38)(34,36)(35,44)(37,47)(39,48)(40,43)(41,45)(42,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=g,e^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, 8638089);