#7641800

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,f^2=hi,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, 7548996);