#8927042

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

4290 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^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, 8834238);