#8734931

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,29)(28,32)(33,34,39,43)(35,36,46,40)(37,41,38,42)(44,47,45,48);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,35,39,46)(34,40,43,36)(37,44,38,45)(41,48,42,47);(3,11)(4,12)(7,16)(8,13)(17,18,22,26)(19,23,30,27)(20,24,29,31)(21,25,28,32)(33,35)(34,36)(37,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,36)(34,35)(37,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,37,39,38)(34,41,43,42)(35,44,46,45)(36,47,40,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,38,39,37)(34,42,43,41)(35,45,46,44)(36,48,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=gh,f^2=g,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, 8642127);