#8700336

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	#52292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=h,e^2=gh,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, 8607532);