#372416

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

1249 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)(9,10,14,19,15,20,13,18)(11,21,24,16,25,32,23,33)(12,26,28,36,30,35,29,17)(22,31,27,34,38,40,37,39);(9,11)(10,16)(12,27,30,37)(13,24)(14,23)(15,25)(17,34,36,39)(18,32)(19,21)(20,33)(22,29,38,28)(26,31,35,40);(2,5)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,35)(20,36)(21,34)(23,27)(24,37)(25,38)(32,39)(33,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,31)(18,32)(19,21)(20,33)(26,39)(27,29)(28,37)(30,38)(34,35)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279612);