#9913269

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

4024 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^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, 9820465);