#205715

order	512
nilpotency class	3
isoclinism	isoclinic class of #205565
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112911);