#8159032

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

4072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066228);