#8902500

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

4183 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(33,34,40,42)(35,36,38,37)(39,47,48,41)(43,45,46,44);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,47)(46,48);(2,6)(8,9)(10,13)(15,16)(18,23)(21,30)(24,31)(28,32)(33,36,40,37)(34,35,42,38)(39,45,48,44)(41,46,47,43);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,37,40,36)(34,35,42,38)(39,45,48,44)(41,43,47,46);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32)(33,38,40,35)(34,37,42,36)(39,46,48,43)(41,45,47,44);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,39)(34,41)(35,43)(36,44)(37,45)(38,46)(40,48)(42,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8809696);