#420411

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

1818 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=e,e^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,e^b=ei,f^b=fi,h^b=hi,f^c=fi,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	#72238740, #72238741

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327607);