#8467003

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	#42508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8374199);