#8348937

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

4253 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)(13,14,15,16)(17,18,23,27)(19,24,28,32)(20,21,25,30)(22,26,31,29)(34,40)(36,41)(42,47)(44,48);(2,7)(3,12)(4,10)(14,16)(17,19)(18,24)(20,22)(21,26)(23,28)(25,31)(27,32)(29,30)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(2,7)(5,9)(8,11)(14,16)(19,28)(22,31)(24,32)(26,29)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,3)(2,8)(4,15)(5,14)(6,12)(7,11)(9,16)(10,13)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,31)(29,32)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,4,6,10)(2,9,7,5)(3,13,12,15)(8,14,11,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,30)(26,28)(31,32)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,5,6,9)(2,10,7,4)(3,14,12,16)(8,13,11,15)(17,22)(18,26)(19,20)(21,24)(23,31)(25,28)(27,29)(30,32)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8256133);