#8701358

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 8608554);