#8497760

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	#53446	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404956);