#9851095

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	#49812	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=gh,e^2=i,f^2=g,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, 9758291);