#455967

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×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1155 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,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, 363163);