#7054090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308420, #72308526, #72308535
maximal normal subgroup of	#72308420, #72308526, #72308535

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961286);