#372789

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1481 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:

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

Polycyclic presentation:

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

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, 279985);