#110585

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

622 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17781);