#8096382

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,15)(14,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,38,42)(35,39,44,47)(36,40,43,46)(37,41,45,48);(2,7)(3,12)(9,16)(10,13)(20,28)(25,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,14)(15,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,38,44)(34,39,42,47)(36,37,43,45)(40,41,46,48);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,14)(15,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,9)(3,13)(5,8)(6,14)(7,16)(10,12)(11,15)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,36)(34,40)(35,37)(38,43)(39,41)(42,46)(44,45)(47,48);(1,5)(2,10)(3,9)(4,11)(6,15)(7,13)(8,14)(12,16)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,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, 8003578);