#8749997

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	#45002	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=gh,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, 8657193);