#7703097

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,33)(27,31,29,38)(28,35,37,32)(34,40,36,39);(2,5)(6,8)(25,27,30,29)(26,31,33,38)(28,36,37,34)(32,40,35,39);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,28,30,37)(26,32,33,35)(27,34,29,36)(31,39,38,40);(11,18)(14,17)(19,24)(22,23)(25,29,30,27)(26,31,33,38)(28,36,37,34)(32,39,35,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,33)(28,37)(31,38)(34,36);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=hi,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, 7610293);