#9913331

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

3976 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^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, 9820527);