#110876

order	512
nilpotency class	3
isoclinism	isoclinic class of #110864
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

593 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:

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

Polycyclic presentation:

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

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, 18072);