#8517624

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

4322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424820);