#8806870

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

4063 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714066);