#210400

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#18984	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

862 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117596);