#8202196

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	#38505	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:

(1,2,7,9)(3,8,12,16)(4,6,11,14)(5,13,15,10)(17,18,22,27)(19,26,29,23)(20,25,21,24)(28,31,30,32)(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)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,32)(26,27)(35,43)(36,46)(39,48)(40,45);(2,9)(3,12)(6,14)(10,13)(17,20,22,21)(18,24,27,25)(19,28,29,30)(23,31,26,32)(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)(17,18)(19,26)(20,24)(21,25)(22,27)(23,29)(28,32)(30,31);(1,5)(2,10)(3,6)(4,8)(7,15)(9,13)(11,16)(12,14)(17,21,22,20)(18,25,27,24)(19,28,29,30)(23,31,26,32)(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)(17,19)(18,26)(20,30)(21,28)(22,29)(23,27)(24,31)(25,32)(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=gh,c^2=g,d^2=gh,e^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, 8109392);