#6576174

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2211 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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	#72256956, #72256988, #72257004, …
maximal normal subgroup of	#72256956, #72256988, #72257004, …
maximal quotient of	#72264214, #72265037, #72265506

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483370);