#8211666

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	#37912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=ghi,e^2=gh,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, 8118862);