#9712158

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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)(19,26)(20,25)(21,30)(22,23)(24,29)(27,28)(31,32);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(18,23)(20,30)(24,26)(27,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(2,4)(7,9)(8,12)(14,15)(17,19,22,29)(18,24,23,26)(20,27,30,32)(21,28,25,31)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,22,30)(18,25,23,21)(19,27,29,32)(24,31,26,28)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,27,26,32)(33,37,38,34)(35,41,43,39)(36,40,44,45)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=hi,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, 9619354);