#8695237

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

3937 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602433);