#8534821

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

4387 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442017);