#8158940

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3924 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(34,39)(35,42)(41,44)(43,48);(2,8)(4,9)(10,14)(12,16)(17,19,22,20)(18,24,26,23)(21,27,30,28)(25,32,29,31)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20,22,19)(18,23,26,24)(21,28,30,27)(25,31,29,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,7,15)(2,10,8,14)(3,11,5,13)(4,12,9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066136);