#109079

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,14,19,15,18,13,28)(11,20,22,31,23,16,29,32)(12,24,26,33,27,39,25,17)(21,35,36,34,37,38,40,30);(1,2,4,7,5,8,3,6)(9,11)(10,16)(12,21)(13,22)(14,29)(15,23)(17,30)(18,20)(19,31)(24,38)(25,36)(26,40)(27,37)(28,32)(33,34)(35,39);(9,12,13,25,15,27,14,26)(10,17,19,24,18,33,28,39)(11,21,22,36,23,37,29,40)(16,30,31,38,20,34,32,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224831, #72224832, #72224837, …
maximal normal subgroup of	#72224831, #72224832, #72224837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16275);