#8818086

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	#2678	C₂²
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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8725282);