#6583372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14)(15,21)(16,26)(17,27)(18,20)(19,24)(22,23)(25,28);(1,2,3,4)(13,15,18,25)(14,19,20,26)(16,23,24,17)(21,22,28,27);(5,7,8,11)(6,10,9,12)(14,20)(15,16)(17,23)(19,21)(24,25)(26,28);(13,16)(14,21)(15,23)(17,25)(18,24)(19,22)(20,28)(26,27)

Polycyclic presentation:

2^9|c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263890, #72263937, #72264000
maximal normal subgroup of	#72263890, #72263937, #72264000
maximal quotient of	#72264237, #72264306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490568);