#8271928

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8179124);