#355492

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6,9,13,10,7,11,15)(8,14,17,20,16,12,18,19);(1,2,3,4)(6,7)(9,11)(12,14)(17,18);(5,7,9,13,10,6,11,15)(8,14,18,20,16,12,17,19);(5,8)(6,12)(7,14)(9,18)(10,16)(11,17)(13,20)(15,19)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=ce,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi

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, 262688);