#8092469

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
prime radical	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=g,d^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, 7999665);