#10320383

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

3841 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)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(17,18,23,22)(19,24,21,26)(20,27,30,25)(28,32,29,31)(33,34,39,37)(35,42,41,38)(36,44,47,40)(43,46,45,48);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,24)(20,29)(21,23)(22,26)(25,32)(27,31)(28,30)(33,35,39,41)(34,38,37,42)(36,45,47,43)(40,48,44,46);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,20)(18,25)(19,28)(21,29)(22,27)(23,30)(24,31)(26,32)(33,34,39,37)(35,38,41,42)(36,44,47,40)(43,48,45,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,21,23,19)(18,26,22,24)(20,29,30,28)(25,32,27,31)(33,36,39,47)(34,40,37,44)(35,43,41,45)(38,46,42,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,27)(19,29)(21,28)(22,25)(23,30)(24,31)(26,32)(33,37,39,34)(35,38,41,42)(36,40,47,44)(43,48,45,46);(1,4,5,12)(2,7,8,11)(3,9,10,16)(6,13,14,15)(17,22,23,18)(19,26,21,24)(20,27,30,25)(28,32,29,31)(33,38)(34,41)(35,37)(36,46)(39,42)(40,45)(43,44)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=gi,e^2=g,f^2=h,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, 10227579);