#208537

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

918 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)(3,6)(4,5)(7,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,23,24,21)(25,26,30,33,29,32,38,27)(28,35,37,34,36,31,39,40);(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,23)(20,24)(25,27,30,32,29,33,38,26)(28,34,37,35,36,40,39,31);(2,5)(6,8)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,28,29,36)(26,31,32,35)(27,34,33,40)(30,37,38,39);(1,3,4,7)(2,6,5,8)(28,36)(31,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=fh,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115733);