#8295573

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8202769);