#372417

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

1239 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)(9,10,14,20,15,21,13,19)(11,22,25,16,26,33,24,18)(12,27,29,37,31,36,30,17)(23,32,28,34,39,35,38,40);(9,11)(10,16)(12,28,31,38)(13,25)(14,24)(15,26)(17,34,37,40)(18,21)(19,33)(20,22)(23,30,39,29)(27,32,36,35);(2,5)(6,8)(9,12)(10,17)(11,23)(13,29)(14,30)(15,31)(16,32)(18,35)(19,27)(20,36)(21,37)(22,34)(24,28)(25,38)(26,39)(33,40);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,23)(13,25)(14,24)(15,26)(16,21)(17,35)(19,22)(20,33)(27,34)(28,30)(29,38)(31,39)(32,37)(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=dei,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279613);