#472530

order	512
nilpotency class	3
isoclinism	isoclinic class of #457579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1109 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)(7,13,15,10)(8,11)(9,12)(14,18,16,19)(17,20);(1,2,3,4)(5,7)(6,10)(8,16)(9,15)(11,19)(12,13)(14,17)(18,20);(2,4)(5,8)(6,11)(7,14,15,16)(9,17)(10,18,13,19)(12,20);(5,7)(6,10)(8,16)(9,15)(11,18)(12,13)(14,17)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379726);