#9935863

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

4173 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9843059);