#10407799

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	#46768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10314995);