#8105902

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
prime radical	#53182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4348 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)(11,19)(12,16)(13,18)(14,17)(15,22)(20,24)(21,23)(25,26)(27,34)(28,37)(29,32)(30,33)(31,36)(35,39)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,34)(37,40);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,18)(11,20,22,21)(15,23,19,24)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,34,40);(10,17)(15,19)(16,18)(23,24)(26,33)(27,36)(32,37)(35,38);(2,6)(5,7)(9,12,14,13)(10,18,17,16)(11,21,22,20)(15,23,19,24)(25,29,30,28)(26,32,33,37)(27,35,36,38)(31,40,34,39);(9,13,14,12)(10,18,17,16)(11,21,22,20)(15,24,19,23)

Polycyclic presentation:

2^9|c^2=gi,e^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, 8013098);