#9937711

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

4121 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)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,46)(47,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,38)(34,40,43,44)(36,47,42,45)(37,48,41,46);(2,3)(7,9)(8,10)(14,15)(18,20)(24,26)(25,28)(31,32);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,48,44,46);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,47,44,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

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

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, 9844907);