#8193751

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

4119 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,e^2=ghi,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, 8100947);