#8110416

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	#38780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(21,29)(26,30)(27,32)(28,31)(33,34,38,42)(35,36,40,39)(37,45,46,41)(43,47,48,44);(2,7)(3,10)(9,12)(11,16)(18,23)(20,24)(26,30)(28,31)(35,40)(36,39)(37,46)(41,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017612);