#8291632

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

4118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=i,e^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, 8198828);