#8202194

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	#38500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 8109390);