#8183771

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	#45642	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34)(35,46)(36,39)(37,44)(38,43)(40,45)(41,48)(42,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35)(34,41)(36,37)(38,47)(39,43)(40,48)(42,44)(45,46);(2,6)(7,13)(8,11)(14,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36,40,47)(34,42,45,39)(35,37,48,38)(41,44,46,43);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,37,40,38)(34,43,45,44)(35,36,48,47)(39,46,42,41);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,38,40,37)(34,44,45,43)(35,47,48,36)(39,41,42,46);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,39,40,42)(34,36,45,47)(35,43,48,44)(37,46,38,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=g,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, 8090967);