#9910059

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

4313 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=g,f^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, 9817255);