#8183758

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=h,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, 8090954);