#9910088

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

4227 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)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(19,25)(22,29)(26,31)(30,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,9)(3,4)(10,16)(11,12)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,39,36)(34,41,37,40)(38,43,46,44)(42,48,45,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,39,35)(34,40,37,41)(38,44,46,43)(42,47,45,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,37,39,34)(35,40,36,41)(38,42,46,45)(43,48,44,47);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817284);