#7641490

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7548686);