#8621662

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	#45914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,22,24)(19,23,27,28)(20,25,30,32)(21,26,31,29)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,5)(8,10)(9,12)(15,16)(18,24)(19,27)(25,32)(26,29)(35,40)(37,45)(43,46)(44,48);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,20,22,30)(18,25,24,32)(19,26,27,29)(21,23,31,28)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,31)(30,32)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21)(18,26)(19,25)(20,28)(22,31)(23,30)(24,29)(27,32)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=gh,e^2=hi,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, 8528858);