#205669

order	512
nilpotency class	3
isoclinism	isoclinic class of #205546
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

961 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)(3,7)(6,8)(9,10,14,19,15,18,13,27)(11,16,22,32,21,30,28,31)(12,23,25,33,26,37,24,17)(20,34,36,29,35,40,38,39);(1,2,3,5,4,6,7,8)(9,11,14,28,15,21,13,22)(10,16,27,32,18,30,19,31)(12,20,25,38,26,35,24,36)(17,29,37,40,33,39,23,34);(9,12,14,25,15,26,13,24)(10,17,27,37,18,33,19,23)(11,20,28,38,21,35,22,36)(16,29,32,40,30,39,31,34);(2,6)(5,8)(11,21)(16,30)(20,35)(22,28)(29,39)(31,32)(34,40)(36,38)

Polycyclic presentation:

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

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, 112865);