#4100158

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gh,c^2=fghi,d^2=gh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4007354);