#8186913

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	#44975	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(2,9)(6,10)(8,11)(13,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094109);