#8197098

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104294);