#137616

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44812);