#9712013

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,18,23,25)(19,24,28,22)(20,26,29,32)(21,27,31,30)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,23,28)(18,24,25,22)(20,21,29,31)(26,27,32,30)(36,44)(40,47)(42,48)(45,46);(18,25)(19,28)(20,29)(27,30)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,20)(18,26)(19,21)(22,30)(23,29)(24,27)(25,32)(28,31)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,5,7,2)(3,9,4,8)(6,10,14,13)(11,15,12,16)(17,21)(18,27)(19,20)(22,32)(23,31)(24,26)(25,30)(28,29)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,24)(18,28,25,19)(20,30,29,27)(21,26,31,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619209);