#109299

order	512
nilpotency class	3
isoclinism	isoclinic class of #109288
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16495);