#8745834

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3826 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=ghi,f^2=gh,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, 8653030);