#10316827

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	#2815	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=gh,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, 10224023);