#10320321

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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)(3,10)(4,8)(5,6)(7,9)(11,14)(12,13)(15,16)(17,18)(19,28)(20,24)(21,32)(22,27)(23,30)(25,31)(26,29)(33,34,39,38)(35,42,37,40)(36,44,46,41)(43,47,45,48);(3,4)(8,10)(11,13)(12,14)(17,19)(18,23)(20,21)(22,30)(24,26)(25,32)(27,28)(29,31)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,3,7,4)(2,8,9,10)(5,11,15,13)(6,12,16,14)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,26,28,32)(33,35,39,37)(34,40,38,42)(36,43,46,45)(41,47,44,48);(1,4,7,3)(2,8,9,10)(5,11,15,13)(6,14,16,12)(17,20)(18,25)(19,29)(21,30)(22,31)(23,26)(24,27)(28,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,5,7,15)(2,6,9,16)(3,11,4,13)(8,12,10,14)(17,18)(19,28)(20,24)(21,32)(22,27)(23,30)(25,31)(26,29)(33,37,39,35)(34,40,38,42)(36,43,46,45)(41,48,44,47);(1,6)(2,5)(3,12)(4,14)(7,16)(8,11)(9,15)(10,13)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,24,28,25)(33,38,39,34)(35,42,37,40)(36,44,46,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

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, 10227517);