#116019

order	512
nilpotency class	3
isoclinism	isoclinic class of #116016
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23215);