#8504782

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411978);