#8095268

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002464);