#375504

order	512
nilpotency class	3
isoclinism	isoclinic class of #213778
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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)(3,7)(4,6)(5,8)(9,10)(11,19,22,15)(12,23,26,38)(13,18)(14,17)(16,27,31,40)(20,35)(21,33,28,30)(24,37)(25,32)(29,36)(34,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,24)(13,21)(14,22)(16,29)(17,19)(18,30)(20,26)(23,37)(25,28)(27,39)(31,34)(32,33)(35,38)(36,40);(9,12,25,40,14,26,13,27)(10,16,18,23,17,31,32,38)(11,20,21,36,22,24,28,39)(15,29,33,35,19,34,30,37);(2,6)(5,7)(9,11)(10,15)(12,20)(13,28)(14,22)(16,29)(17,19)(18,33)(21,25)(23,35)(24,26)(27,39)(30,32)(31,34)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fghi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,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, 282700);