#8165572

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	#40701	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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8072768);