#190138

order	512
nilpotency class	3
isoclinism	isoclinic class of #190107
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1772 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)(3,7,8,5)(4,6)(9,10,13,18,14,17,27,32)(11,19,22,33,21,15,26,29)(12,23,25,37,24,16,30,31)(20,28,36,39,35,34,38,40);(9,11)(10,15)(12,20)(13,26)(14,21)(16,28)(17,19)(18,33)(22,27)(23,34)(24,35)(25,38)(29,32)(30,36)(31,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16,17,23)(11,20)(13,25)(14,24)(15,28,19,34)(18,31,32,37)(21,35)(22,36)(26,38)(27,30)(29,39,33,40);(11,21)(15,19)(20,35)(22,26)(28,34)(29,33)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,c^2=e,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fh,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, 97334);