#8752009

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

4173 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8659205);