#8096772

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

4094 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003968);