#8285462

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	#40825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8192658);