#8607432

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=g,f^2=h,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, 8514628);