#8186998

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094194);