#8932311

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

4125 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=gi,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, 8839507);