#457596

order	512
nilpotency class	3
isoclinism	isoclinic class of #453991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1521 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)(11,23)(12,19)(13,31)(14,21)(15,20)(16,22)(17,26)(18,30)(24,32)(25,36)(27,35)(28,34)(29,37)(33,39)(38,40);(2,5)(6,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,26)(16,28)(18,36)(19,33)(20,34)(21,23)(22,35)(29,40)(30,39)(31,38)(32,37);(1,3)(2,6)(4,7)(5,8)(9,12,15,30,16,13,14,29)(10,18,21,19,22,37,20,31)(11,24,27,40,28,25,26,39)(17,32,34,33,35,36,23,38);(3,7)(6,8)(9,13,16,12)(10,19,22,31)(11,25,28,24)(14,29,15,30)(17,33,35,38)(18,20,37,21)(23,36,34,32)(26,39,27,40)

Polycyclic presentation:

2^9|a^2=ef,c^2=f,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,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, 364792);