#8125491

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	#57972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=i,e^2=h,f^2=ghi,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, 8032687);