#445740

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=g,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338910, #72338922, #72338924
maximal normal subgroup of	#72338910, #72338922, #72338924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352936);