#110068

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

776 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17264);