#9923899

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

4167 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,25)(21,30)(22,24)(23,28)(27,32)(29,31)(34,38)(36,44)(39,41)(42,48);(1,3,7,4)(2,8,10,5)(6,11,15,12)(9,14,16,13)(17,19)(18,23)(20,27)(21,25)(22,28)(24,26)(29,30)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(18,24)(20,29)(23,26)(27,30)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,7,3)(2,8,10,5)(6,12,15,11)(9,14,16,13)(17,20,22,29)(18,25,24,31)(19,27,28,30)(21,26,32,23)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,7,8)(2,3,10,4)(6,14,15,13)(9,12,16,11)(17,19)(18,26)(20,27)(21,31)(22,28)(23,24)(25,32)(29,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,14)(10,16)(17,21,22,32)(18,27,24,30)(19,25,28,31)(20,26,29,23)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gi,e^2=gh,f^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, 9831095);