#8618104

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	#41249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8525300);