#9909994

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

3884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^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, 9817190);