#8092472

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	#52394	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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)(9,10,14,18)(11,17,13,15)(12,16,20,23)(19,24,21,22)(25,26,30,33)(27,34,37,31)(28,38,29,32)(35,39,36,40);(3,7)(5,8)(9,11,14,13)(10,15,18,17)(12,19,20,21)(16,22,23,24)(25,27,30,37)(26,31,33,34)(28,35,29,36)(32,39,38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,20)(10,16,18,23)(11,19,13,21)(15,22,17,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40);(3,7)(5,8)(9,13,14,11)(10,17,18,15)(12,19,20,21)(16,22,23,24)(25,28,30,29)(26,32,33,38)(27,36,37,35)(31,40,34,39);(2,6)(5,8)(10,18)(15,17)(16,23)(22,24);(25,29,30,28)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 7999668);