#9712811

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

3733 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=gi,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, 9620007);