#471051

order	512
nilpotency class	3
isoclinism	isoclinic class of #460096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
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

960 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:

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

Polycyclic presentation:

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

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, 378247);