#8192505

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=h,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, 8099701);