#8687328

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	#58129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594524);