#474779

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
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

848 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381975);