#9874905

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	#40145	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(3,9)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,7)(4,13)(8,9)(10,16)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(18,23)(22,29)(24,27)(30,31)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,20)(18,24)(19,25)(21,28)(22,30)(23,27)(26,32)(29,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=gi,e^2=gi,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, 9782101);