#8096397

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003593);