#8105900

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,30,34)(27,35,37,31)(28,33,29,32)(36,39,38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,39,35,40);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,30,28)(26,33,34,32)(27,36,37,38)(31,39,35,40);(2,6)(5,7)(10,18)(11,21)(16,17)(20,22)(27,37)(31,35)(36,38)(39,40);(9,13,14,12)(10,17,18,16)(11,22,21,20)(15,24,19,23)

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi

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, 8013096);