#130016

order	512
nilpotency class	4
isoclinism	isoclinic class of #129789
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

577 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37212);