#190178

order	512
nilpotency class	3
isoclinism	isoclinic class of #190107
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,g^2=i,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, 97374);