#8595737

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	#38013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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)(3,7,10,14)(4,5,8,13)(11,12,15,16)(17,18,23,27)(19,24,30,29)(20,31,21,25)(22,26,32,28)(33,34,38,39)(35,42,37,40)(36,44,45,41)(43,47,46,48);(1,3,6,10)(2,7,9,14)(4,11,8,15)(5,12,13,16)(17,19)(18,24)(20,26)(21,28)(22,25)(23,30)(27,29)(31,32)(34,39)(36,45)(40,42)(43,46);(3,10)(7,14)(11,15)(12,16)(17,20,23,21)(18,25,27,31)(19,28,30,26)(22,24,32,29)(33,35,38,37)(34,40,39,42)(36,46,45,43)(41,48,44,47);(1,4)(2,5)(3,11)(6,8)(7,12)(9,13)(10,15)(14,16)(17,21,23,20)(18,25,27,31)(19,26,30,28)(22,24,32,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,18)(19,29)(20,31)(21,25)(22,26)(23,27)(24,30)(28,32)(33,36,38,45)(34,41,39,44)(35,43,37,46)(40,47,42,48);(1,5)(2,8)(3,12)(4,9)(6,13)(7,15)(10,16)(11,14)(17,22)(18,26)(19,25)(20,24)(21,29)(23,32)(27,28)(30,31)(33,37,38,35)(34,40,39,42)(36,46,45,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=g,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, 8502933);