#10368657

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

4267 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,46,48);(2,3)(5,13)(7,9)(12,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35,39,40)(34,37,36,42)(38,46,47,43)(41,48,44,45);(4,11)(7,9)(12,16)(14,15)(18,20)(23,25)(24,28)(31,32)(33,34,39,36)(35,42,40,37)(38,41,47,44)(43,48,46,45);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,39,34)(35,37,40,42)(38,44,47,41)(43,45,46,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,37)(34,40)(35,36)(38,45)(39,42)(41,46)(43,44)(47,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275853);