#8159232

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

3914 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)(8,14)(9,10)(11,16)(13,15)(17,18,22,23)(19,25,20,24)(21,29,30,26)(27,31,28,32)(35,36)(39,40)(43,44)(46,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(18,23)(24,25)(26,29)(31,32)(33,34,38,42)(35,39,36,40)(37,41,45,48)(43,46,44,47);(2,7)(3,9)(8,12)(11,15)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(1,2)(3,10)(4,12)(5,9)(6,7)(8,14)(11,13)(15,16)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,31)(29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,14)(2,8,7,12)(3,11,9,15)(5,13,10,16)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,5,6,10)(2,9,7,3)(4,13,14,16)(8,15,12,11)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,e^2=h,f^2=gh,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, 8066428);