#8295568

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

4066 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,34,38,39)(35,37,41,46)(36,40,45,47)(42,44,43,48);(2,8)(5,16)(7,11)(9,14)(21,29)(24,31)(27,28)(30,32)(34,39)(37,46)(40,47)(44,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,15)(14,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,35,38,41)(34,37,39,46)(36,43,45,42)(40,48,47,44);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,43)(46,48);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,43)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202764);