#8932241

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
prime radical	#47010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(19,26)(21,29)(23,24)(25,28)(34,36)(35,42)(38,46)(45,47);(1,3,7,11)(2,8,10,15)(4,12,13,5)(6,9,14,16)(17,18,22,20)(19,24,26,23)(21,25,29,28)(27,32,30,31)(33,34,39,36)(35,37,42,40)(38,41,46,44)(43,45,48,47);(3,11)(4,13)(6,14)(8,15)(17,19,22,26)(18,23,20,24)(21,30,29,27)(25,32,28,31)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(1,4)(2,6)(3,12)(5,11)(7,13)(8,9)(10,14)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,32)(28,31)(29,30)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,5,7,12)(2,9,10,16)(3,13,11,4)(6,15,14,8)(17,20,22,18)(19,24,26,23)(21,28,29,25)(27,32,30,31)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,6)(2,4)(3,9)(5,15)(7,14)(8,12)(10,13)(11,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,31)(24,32)(26,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=g,e^2=gi,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, 8839437);