#477157

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384353);