#8341620

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

4074 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)(3,5,10,8)(4,12,15,9)(11,16,13,14)(18,23)(19,24)(25,30)(26,31)(33,34,39,37)(35,38,41,43)(36,40,45,46)(42,44,47,48);(2,7)(3,10)(9,12)(11,13)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,44)(45,47)(46,48);(2,7)(5,8)(9,12)(14,16)(21,29)(25,30)(26,31)(27,32)(35,41)(38,43)(42,47)(44,48);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,45)(41,47)(43,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37,39,34)(35,43,41,38)(36,40,45,46)(42,44,47,48);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,38,39,43)(34,41,37,35)(36,44,45,48)(40,47,46,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=hi,f^2=hi,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, 8248816);