#8285404

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	#49722	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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:

(1,2,7,10)(3,8,11,6)(4,5,9,15)(12,13,16,14)(17,18)(19,27)(20,22)(21,25)(23,24)(26,30)(28,29)(31,32)(35,37)(39,42)(43,46)(47,48);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(18,24)(20,30)(25,27)(28,31)(33,34)(35,42)(36,44)(37,39)(38,41)(40,45)(43,47)(46,48);(3,11)(6,8)(12,16)(13,14)(17,19,23,21)(18,25,24,27)(20,28,30,31)(22,32,26,29)(33,35,38,37)(34,39,41,42)(36,43,45,46)(40,47,44,48);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,29)(27,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,44)(42,48);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,21,23,19)(18,27,24,25)(20,28,30,31)(22,32,26,29)(33,37,38,35)(34,39,41,42)(36,43,45,46)(40,48,44,47);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,22,23,26)(18,20,24,30)(19,29,21,32)(25,31,27,28)(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=ghi,c^2=h,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, 8192600);