#115994

order	512
nilpotency class	3
isoclinism	isoclinic class of #111436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

606 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:

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

Polycyclic presentation:

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

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, 23190);