#409347

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,d^2=ei,e^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,f^b=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224752, #72224756, #72224758
maximal normal subgroup of	#72224752, #72224756, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316543);