#190126

order	512
nilpotency class	3
isoclinism	isoclinic class of #190124
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^b=gh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97322);