#190226

order	512
nilpotency class	3
isoclinism	isoclinic class of #190224
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1079 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)(4,7,12,14)(5,8)(10,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,27)(18,22,24,30)(20,26,28,32)(23,29,25,31);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(2,8)(3,11)(7,14)(10,16)(18,24)(22,30)(23,25)(29,31)

Polycyclic presentation:

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

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, 97422);