#10320369

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	#43418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,25)(20,31)(21,22)(23,26)(24,27)(28,30)(29,32)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,24,23)(18,25,27,26)(20,29,21,28)(22,30,31,32)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(2,3)(7,9)(8,10)(14,15)(17,20,24,21)(18,22,27,31)(19,28,23,29)(25,32,26,30)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,24,20)(18,22,27,31)(19,29,23,28)(25,32,26,30)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,22)(18,21)(19,30)(20,27)(23,32)(24,31)(25,28)(26,29)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,23,24,19)(18,26,27,25)(20,29,21,28)(22,30,31,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=i,f^2=gh,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, 10227565);