#8434131

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	#52372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341327);