#8757417

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

3767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664613);