#8285414

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

3903 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)(3,11)(4,14)(5,8)(6,12)(7,9)(10,15)(13,16)(17,18,23,26)(19,24,27,22)(20,21,25,31)(28,29,32,30)(35,37)(39,42)(43,46)(47,48);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,14,16,10)(17,19,23,27)(18,24,26,22)(20,28,25,32)(21,29,31,30)(33,34)(35,42)(36,44)(37,39)(38,41)(40,45)(43,47)(46,48);(2,9)(6,16)(8,11)(10,14)(19,27)(22,24)(28,32)(29,30)(33,35,38,37)(34,39,41,42)(36,43,45,46)(40,47,44,48);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,31)(27,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,44)(42,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,21)(18,25)(19,29)(20,26)(22,28)(23,31)(24,32)(27,30)(33,37,38,35)(34,39,41,42)(36,43,45,46)(40,48,44,47);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,22)(18,19)(20,30)(21,28)(23,24)(25,29)(26,27)(31,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8192610);