#8295499

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

4085 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(18,23)(19,24)(25,30)(26,31);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,34,38,39)(35,37,41,46)(36,40,45,47)(42,44,43,48);(2,9)(6,10)(8,11)(13,16)(21,29)(25,30)(26,31)(27,32)(34,39)(37,46)(40,47)(44,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,13)(11,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,35,38,41)(34,37,39,46)(36,43,45,42)(40,48,47,44);(1,5,7,3)(2,11,9,8)(4,12,15,14)(6,16,10,13)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,43)(46,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,43)(42,47)(45,48)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=h,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, 8202695);