#8193742

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,e^2=hi,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, 8100938);