#8095261

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

4158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=gh,e^2=i,f^2=g,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, 8002457);