#10316822

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	#52707	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=gh,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, 10224018);