#8159142

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=g,e^2=h,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, 8066338);