#8227427

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	#56294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8134623);