#8158882

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	#41825	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=h,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, 8066078);