#8159010

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	#41638	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,4)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,18,22,21)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066206);