#8701364

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,6)(4,10)(5,16)(7,11)(8,12)(9,14)(13,15)(34,36)(35,41)(40,44)(43,47);(1,3)(2,9)(4,13)(5,7)(6,12)(8,14)(10,11)(15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,4,8,5)(2,10,12,16)(3,13,14,7)(6,15,9,11)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,8,4)(2,10,12,16)(3,13,14,7)(6,11,9,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,6,8,9)(2,3,12,14)(4,15,5,11)(7,10,13,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,7,8,13)(2,11,12,15)(3,5,14,4)(6,16,9,10)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608560);