#214904

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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

987 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,f^2=i,g^2=h,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, 122100);