#411040

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

761 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:

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

Polycyclic presentation:

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

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, 318236);