#8092488

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

4517 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)(11,17,13,15)(12,16,21,24)(19,23,20,22)(25,26,30,34)(27,35,38,31)(28,33,29,32)(36,39,37,40);(3,7)(5,8)(9,11,14,13)(10,15,18,17)(12,19,21,20)(16,22,24,23)(25,27,30,38)(26,31,34,35)(28,36,29,37)(32,39,33,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,21)(10,16,18,24)(11,19,13,20)(15,22,17,23)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(3,7)(5,8)(9,11,14,13)(10,15,18,17)(12,20,21,19)(16,23,24,22)(25,28,30,29)(26,32,34,33)(27,37,38,36)(31,40,35,39);(2,6)(5,8)(9,13,14,11)(10,17,18,15)(12,20,21,19)(16,23,24,22)(25,28,30,29)(26,33,34,32)(27,36,38,37)(31,40,35,39);(25,29,30,28)(26,32,34,33)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

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, 7999684);