#8107239

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	#41261	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8014435);