#8818035

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	#46505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=hi,e^2=g,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, 8725231);