#420407

order	512
nilpotency class	3
isoclinism	isoclinic class of #420407
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	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2084 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=e,f^2=i,h^2=i,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,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256726, #72256729, #72256732
maximal normal subgroup of	#72256726, #72256729, #72256732
maximal quotient of	#72258188

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327603);