#8757354

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	#58558	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=g,e^2=gi,f^2=gh,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, 8664550);