#8630912

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

4026 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,22,25)(19,21,24,31)(20,27,30,23)(26,32,28,29)(33,34,38,37)(35,41,43,39)(36,44,46,40)(42,47,45,48);(2,7)(3,10)(9,12)(11,16)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,35,38,43)(34,39,37,41)(36,42,46,45)(40,47,44,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,21)(20,26)(22,24)(23,29)(25,31)(27,32)(28,30)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,48)(45,46);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,37,38,34)(35,41,43,39)(36,40,46,44)(42,47,45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,e^2=ghi,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, 8538108);