#8922910

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

4224 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8830106);