#9712868

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3748 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9620064);