#8095410

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

3920 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(37,45)(40,44)(42,48)(46,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(6,13)(8,11)(14,15)(20,28)(23,30)(26,32)(29,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=g,e^2=gi,f^2=h,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, 8002606);