#8728487

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

4161 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,25)(19,27,21,24)(20,26,31,30)(22,28,32,29)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,30,32,26)(33,35,38,45)(34,39,42,43)(36,37,44,47)(40,46,48,41);(3,10)(6,7)(11,16)(14,15)(18,25)(20,31)(24,27)(28,29)(33,36,38,44)(34,40,42,48)(35,37,45,47)(39,46,43,41);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27)(33,36)(34,40)(35,37)(38,44)(39,46)(41,43)(42,48)(45,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,19)(18,27,25,24)(20,28,31,29)(22,30,32,26)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,45)(44,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22)(18,28)(19,30)(20,27)(21,26)(23,32)(24,31)(25,29)(33,37)(34,41)(35,44)(36,45)(38,47)(39,40)(42,46)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635683);