#8445759

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

4015 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,e^2=gi,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, 8352955);