#8924698

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	#57184	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831894);