#8167421

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

4056 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)(8,10)(11,16)(14,15)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(2,3)(7,9)(8,10)(14,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,34,38,40)(35,36,43,41)(37,39,46,45)(42,44,48,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36,38,41)(34,35,40,43)(37,44,46,47)(39,42,45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gi,e^2=i,f^2=g,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, 8074617);