#8284508

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	#54814	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,e^2=g,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, 8191704);