#214924

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
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	C₂×C₄²	C₂⁴
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,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, 122120);