#8931111

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

3977 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:

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

Polycyclic presentation:

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

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, 8838307);