#8201969

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

4184 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)(3,9)(4,14)(5,10)(6,11)(7,8)(12,13)(15,16)(17,18,23,27)(19,22,29,31)(20,26,30,24)(21,25,28,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,8)(5,11)(9,14)(12,15)(17,19,23,29)(18,22,27,31)(20,21,30,28)(24,32,26,25)(35,43)(36,46)(39,48)(40,45);(1,3,7,4)(2,9,8,14)(5,12,11,15)(6,16,10,13)(17,20,23,30)(18,24,27,26)(19,28,29,21)(22,25,31,32)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,4,7,3)(2,9,8,14)(5,15,11,12)(6,16,10,13)(17,21)(18,25)(19,20)(22,26)(23,28)(24,31)(27,32)(29,30);(1,5,7,11)(2,10,8,6)(3,12,4,15)(9,13,14,16)(17,22)(18,19)(20,25)(21,26)(23,31)(24,28)(27,29)(30,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,6,7,10)(2,11,8,5)(3,13,4,16)(9,12,14,15)(17,20)(18,26)(19,21)(22,25)(23,30)(24,27)(28,29)(31,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8109165);