#8696224

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603420);