#409344

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 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)(3,6,7,8)(9,10,14,21,16,22,15,20)(11,23,26,17,27,39,29,36)(12,28,30,18,25,40,34,37)(13,31,32,19,33,38,24,35);(2,5)(6,8)(9,11,14,26,16,27,15,29)(10,17,20,23,22,36,21,39)(12,24,30,13,25,32,34,33)(18,19,28,31,37,35,40,38);(9,12)(10,18)(11,24)(13,26)(14,30)(15,34)(16,25)(17,35)(19,36)(20,28)(21,40)(22,37)(23,38)(27,32)(29,33)(31,39);(1,3,4,7)(2,6,5,8)(9,13,14,32,16,33,15,24)(10,19,20,31,22,35,21,38)(11,25,26,34,27,12,29,30)(17,28,23,37,36,40,39,18)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,d^2=e,e^2=i,f^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,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265212
maximal normal subgroup of	#72265212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316540);