#9712228

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	#41270	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,26)(20,25)(21,30)(22,23)(24,29)(27,28)(31,32)(33,34,39,43)(35,40,38,42)(36,37,41,47)(44,48,46,45);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(18,23)(20,30)(24,26)(27,32)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,47,48);(2,8)(4,9)(7,14)(10,15)(17,19,22,29)(18,24,23,26)(20,27,30,32)(21,28,25,31)(33,36,39,41)(34,37,43,47)(35,44,38,46)(40,48,42,45);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,20,22,30)(18,25,23,21)(19,27,29,32)(24,31,26,28)(33,37)(34,41)(35,45)(36,43)(38,48)(39,47)(40,44)(42,46);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,27,26,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,47,48)

Polycyclic presentation:

2^9|a^2=i,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, 9619424);