#8095491

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

4051 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=hi,e^2=gi,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, 8002687);