#215071

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	C₂×C₄²	C₂⁴
prime radical	#35844	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

954 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)(3,6,10,16)(4,9)(5,7)(8,12)(11,14,13,15)(17,18,21,25,22,26,20,24)(19,27,30,23,28,32,29,31);(1,3)(2,6)(4,10)(5,13)(7,15)(8,14)(9,16)(11,12)(19,28)(23,31)(27,32)(29,30);(2,7)(5,12)(6,14)(8,9)(11,13)(15,16)(17,19,22,28)(18,23,26,31)(20,29,21,30)(24,27,25,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,15)(9,16)(12,13)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,27,31,32)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=hi,f^2=i,g^2=hi,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, 122267);