#8824434

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

3850 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8731630);