#8709200

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8616396);