#419793

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1689 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)(7,16,19,17)(8,12,18,13)(9,14)(10,20)(11,15);(1,2)(3,4)(5,7)(6,12)(8,9)(10,18)(11,19)(13,15)(14,17)(16,20);(2,4)(5,6)(7,17)(8,13)(9,14)(10,20)(11,15)(12,18)(16,19);(5,8,10,19,11,18,9,7)(6,13,20,16,15,12,14,17)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238734, #72238735
maximal normal subgroup of	#72238734, #72238735
maximal quotient of	#72238738, #72238745, #72238746

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326989);