#187816

order	512
nilpotency class	3
isoclinism	isoclinic class of #187756
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3212 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95012);