#472680

order	512
nilpotency class	3
isoclinism	isoclinic class of #456884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

782 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:

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

Polycyclic presentation:

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

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, 379876);