#9910724

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

4137 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,32,29)(33,34)(35,41)(36,40)(37,42)(38,39)(43,44)(45,47)(46,48);(3,9)(7,13)(10,15)(14,16)(18,23)(20,27)(25,31)(28,29)(33,34)(35,37)(36,40)(38,39)(41,42)(43,46)(44,48)(45,47);(2,6)(7,13)(8,11)(14,16)(34,39)(35,42)(40,47)(43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,24)(18,20,23,27)(21,26,30,32)(25,28,31,29)(33,35)(34,37)(36,43)(38,42)(39,41)(40,46)(44,47)(45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,24)(19,23)(21,29)(22,27)(25,26)(28,30)(31,32)(33,36,38,45)(34,40,39,47)(35,43,42,48)(37,46,41,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37)(34,35)(36,44)(38,41)(39,42)(40,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=h,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, 9817920);