#415458

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=ef,c^2=ef,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322654);