#477022

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,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,d^b=def,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, 384218);