#8285526

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

4055 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,39,42)(35,40,43,38)(36,37,41,47)(44,45,48,46);(2,7)(3,10)(9,12)(11,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,43)(34,40,42,38)(36,44,41,48)(37,45,47,46);(2,7)(5,8)(9,12)(14,16)(21,29)(24,31)(26,32)(28,30)(35,43)(38,40)(44,48)(45,46);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37)(34,41)(35,45)(36,42)(38,44)(39,47)(40,48)(43,46);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38)(34,35)(36,46)(37,44)(39,40)(41,45)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=gh,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, 8192722);