#363429

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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)(4,5)(6,8)(9,10,14,17,15,19,13,20)(11,21,24,32,25,16,23,30)(12,26,27,33,28,18,29,34)(22,31,36,40,37,35,38,39);(2,5)(6,7)(9,11,15,25)(10,16,19,21)(12,22,28,37)(13,24,14,23)(17,32,20,30)(18,31,26,35)(27,38,29,36)(33,40,34,39);(10,17)(13,14)(16,30)(18,33)(19,20)(21,32)(23,24)(26,34)(27,29)(31,39)(35,40)(36,38);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,22)(13,27)(14,29)(15,28)(16,31)(17,33)(19,26)(20,34)(21,35)(23,36)(24,38)(25,37)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,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, 270625);