#8158965

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

3850 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8066161);