#214765

order	512
nilpotency class	3
isoclinism	isoclinic class of #214736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

989 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)(3,7)(4,11)(5,14)(6,9)(8,10)(12,16)(13,15)(17,18)(19,23,20,25)(21,27)(22,24)(26,30)(28,31,29,32);(1,3,6,5)(2,7,9,14)(4,12,13,10)(8,11,16,15)(17,19,21,20)(18,23,27,25)(22,29,30,28)(24,32,26,31);(1,4,6,13)(2,8,9,16)(3,10,5,12)(7,15,14,11)(18,24)(22,30)(23,31)(25,32)(26,27)(28,29);(17,20,21,19)(18,25,27,23)(22,29,30,28)(24,32,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cei,f^b=fh,g^b=gh,f^c=fi,g^d=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, 121961);