#160413

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=g,c^2=egh,d^2=fg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72237342, #72238090
Fitting subgroup of	#72237342
maximal subgroup of	#72237342, #72238061
maximal normal subgroup of	#72237342, #72238061
maximal quotient of	#72237344

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67609);