#9863089

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3891 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9770285);