#9909972

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

Polycyclic presentation:

2^9|b^2=h,c^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, 9817168);