#8755071

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	#56214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662267);