#120587

order	512
nilpotency class	3
isoclinism	isoclinic class of #120577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

523 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27783);