#232186

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

551 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=de,f^b=fh,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139382);