#8260666

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=h,e^2=ghi,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, 8167862);