#214841

order	512
nilpotency class	3
isoclinism	isoclinic class of #214704
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fi,g^d=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, 122037);