#8097063

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	#45649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(18,24)(20,25)(26,31)(28,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3,6,13)(2,7,9,10)(4,11,8,15)(5,12,14,16)(17,18,23,24)(19,25,21,20)(22,26,30,31)(27,32,29,28)(36,44)(39,46)(43,47)(45,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,23,21)(18,25,24,20)(22,27,30,29)(26,32,31,28)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,4)(2,5)(3,11)(6,8)(7,12)(9,14)(10,16)(13,15)(17,21,23,19)(18,20,24,25)(22,27,30,29)(26,32,31,28)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,5)(2,8)(3,12)(4,9)(6,14)(7,15)(10,11)(13,16)(17,22,23,30)(18,26,24,31)(19,27,21,29)(20,28,25,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=gi,f^2=gi,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, 8004259);