#372422

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 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)(3,6)(4,5)(7,8)(9,10,16,22)(11,23)(12,29,25,35)(13,33)(14,20,15,21)(17,27)(18,31,38,32)(19,30)(24,39)(26,37)(28,36)(34,40);(9,11,15,26,16,28,14,27)(10,17,21,23,22,37,20,36)(12,30,32,13,25,34,31,24)(18,19,35,33,38,40,29,39);(2,5)(6,8)(9,12)(10,18)(11,24)(13,28)(14,31)(15,32)(16,25)(17,33)(19,36)(20,29)(21,35)(22,38)(23,40)(26,30)(27,34)(37,39);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,25)(12,28)(14,30)(15,34)(16,24)(17,35)(18,36)(20,39)(21,33)(22,40)(23,38)(26,31)(27,32)(29,37)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279618);