#420406

order	512
nilpotency class	3
isoclinism	isoclinic class of #420406
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2065 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)(7,15,16,11)(8,12,17,20)(9,18)(10,14)(13,19);(1,2)(3,4)(5,7)(6,11)(8,9)(10,16)(12,13)(14,15)(17,19)(18,20);(2,4)(5,8)(6,12,14,20)(7,9)(10,17)(11,18,15,13)(16,19);(5,9)(6,13,14,18)(7,8)(10,19)(11,20,15,12)(16,17)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327602);