#8621313

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=g,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, 8528509);