#8092482

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

4565 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)(9,10,14,15)(11,16,20,22)(12,18,13,17)(19,24,21,23)(25,26,30,32)(27,35,38,31)(28,34,29,33)(36,39,37,40);(3,7)(5,8)(10,15)(16,22)(17,18)(23,24)(25,27)(26,31)(28,37)(29,36)(30,38)(32,35)(33,40)(34,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23)(26,32)(31,35)(33,34)(39,40);(3,7)(5,8)(9,12,14,13)(10,17,15,18)(11,19,20,21)(16,23,22,24)(25,28,30,29)(26,33,32,34)(27,36,38,37)(31,39,35,40);(2,6)(5,8)(9,13,14,12)(10,18,15,17)(11,19,20,21)(16,23,22,24)(25,28,30,29)(26,34,32,33)(27,37,38,36)(31,39,35,40);(25,29,30,28)(26,33,32,34)(27,36,38,37)(31,40,35,39)

Polycyclic presentation:

2^9|a^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, 7999678);