#8183762

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,18)(19,20)(21,27)(22,24)(23,25)(26,28)(29,32)(30,31)(33,34,38,42)(35,40,36,39)(37,41,45,48)(43,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,19)(18,21)(20,25)(22,26)(23,27)(24,29)(28,31)(30,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,6)(7,13)(8,11)(14,15)(17,20,23,21)(18,19,25,27)(22,28,30,29)(24,26,31,32)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,21,23,20)(18,19,25,27)(22,29,30,28)(24,26,31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,22,23,30)(18,24,25,31)(19,26,27,32)(20,28,21,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,e^2=gh,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, 8090958);