#375688

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

1157 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)(11,19,22,15)(12,23)(13,27)(14,17)(16,26)(18,29)(20,35,24,37)(21,31,28,33)(25,38)(30,36,34,39)(32,40);(2,5)(6,7)(9,11)(10,15)(12,24)(13,21)(14,22)(16,30)(17,19)(18,31)(20,25)(23,37)(26,39)(27,33)(28,29)(32,34)(35,38)(36,40);(9,12,13,26,14,25,29,40)(10,16,18,23,17,32,27,38)(11,20,28,39,22,24,21,36)(15,30,33,35,19,34,31,37);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,20)(13,28)(14,22)(16,30)(17,19)(18,33)(21,29)(23,35)(24,25)(26,39)(27,31)(32,34)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,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, 282884);