#9938089

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
prime radical	#41500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gh,e^2=g,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, 9845285);