#8930253

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=ghi,e^2=h,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, 8837449);