#8211670

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	#35133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8118866);