#8125460

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8032656);