#7703076

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

4474 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(25,26)(27,31)(28,37)(29,32)(30,33)(34,38)(35,40)(36,39);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(27,34)(31,38)(35,36)(39,40);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(25,27,30,34)(26,31,33,38)(28,36,29,35)(32,40,37,39);(12,20)(15,19)(17,24)(21,23)(25,28,30,29)(26,32,33,37)(27,35,34,36)(31,39,38,40);(10,16)(14,22)(15,19)(21,23)(25,29,30,28)(26,32,33,37)(27,36,34,35)(31,39,38,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7610272);