#9712005

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(18,24)(20,28)(23,30)(25,31)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,5,7,2)(3,9,4,8)(6,10,14,13)(11,15,12,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,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, 9619201);