#375506

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1197 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)(9,10,16,21,14,22,15,20)(11,17,24,31,18,12,25,27)(13,19,30,35,28,36,29,34)(23,32,37,40,33,26,38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,22)(13,23)(14,18)(15,24)(16,25)(19,32)(20,31)(21,27)(26,36)(28,33)(29,37)(30,38)(34,40)(35,39);(9,12,14,17)(10,18)(11,22)(13,26,28,32)(15,27,16,31)(19,33)(20,25)(21,24)(23,36)(29,39,30,40)(34,38)(35,37);(2,6)(5,7)(9,13,14,28)(10,19,22,36)(11,23,18,33)(12,26,17,32)(15,30,16,29)(20,35,21,34)(24,38,25,37)(27,40,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fgh,d^2=i,f^2=i,h^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, 282702);