#8689228

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	#38138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4332 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8596424);