#368980

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1162 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)(4,5)(6,8)(9,10,14,18,13,17,25,32)(11,19,22,35,21,15,27,29)(12,16,24,31,23,30,26,33)(20,34,37,40,36,28,38,39);(2,5)(6,7)(9,11,14,22,13,21,25,27)(10,15,18,29,17,19,32,35)(12,20,24,37,23,36,26,38)(16,28,31,39,30,34,33,40);(9,12)(10,16,17,30)(11,20,21,36)(13,23)(14,26)(15,28)(18,33,32,31)(19,34)(22,38,27,37)(24,25)(29,40)(35,39);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,30)(18,32)(28,34)(29,35)(31,33)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,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, 276176);