#374491

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efh,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=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, 281687);