#8749991

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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)(4,10)(5,8)(6,12)(7,9)(13,16)(14,15)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(34,38)(36,43)(39,44)(42,47);(1,3,7,5)(2,8,9,11)(4,15,12,13)(6,14,10,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(35,41)(39,44)(42,47)(46,48);(2,9)(4,12)(8,11)(13,15)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,10)(3,13)(5,15)(6,9)(7,12)(8,14)(11,16)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,5,7,3)(2,11,9,8)(4,15,12,13)(6,14,10,16)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,6,7,10)(2,12,9,4)(3,14,5,16)(8,13,11,15)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657187);