#9910681

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

4062 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,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, 9817877);