#9712229

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	#51119	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=gi,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, 9619425);