#8095795

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

4115 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002991);