#8105628

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#54073	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012824);