#8621640

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

4168 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

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, 8528836);