#8202210

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,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, 8109406);