#8709282

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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)(3,5)(4,12)(6,9)(7,11)(8,13)(10,16)(14,15)(33,34,38,42)(35,43,46,39)(36,45,47,40)(37,41,48,44);(1,3)(2,7)(4,10)(5,9)(6,11)(8,16)(12,14)(13,15)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35,38,46)(34,39,42,43)(36,44,47,41)(37,45,48,40);(4,13)(8,12)(10,15)(14,16)(18,23)(19,25)(24,31)(26,32)(33,35,38,46)(34,39,42,43)(36,44,47,41)(37,45,48,40);(1,4)(2,8)(3,10)(5,14)(6,13)(7,16)(9,12)(11,15)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,35)(34,39)(36,44)(37,40)(38,46)(41,47)(42,43)(45,48);(1,5,6,7)(2,3,9,11)(4,14,13,16)(8,10,12,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,36,38,47)(34,40,42,45)(35,44,46,41)(37,43,48,39);(1,3)(2,5)(4,15)(6,11)(7,9)(8,16)(10,13)(12,14)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,37)(34,41)(35,45)(36,39)(38,48)(40,46)(42,44)(43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8616478);