#7563347

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563346
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213663, #72219565
maximal normal subgroup of	#72213663, #72219565
maximal quotient of	#72226919, #72226931

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470543);