#6676629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2172 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)(7,10)(8,12)(9,11)(14,18)(15,22)(19,21)(23,28);(5,7,10,6)(8,12,9,11)(13,14,16,18)(15,20,22,25)(17,19,24,21)(23,26,28,27);(2,4)(14,19)(15,22)(17,24)(18,21)(20,27)(25,26);(5,8,10,9)(6,11,7,12)(13,15,16,22)(14,20,18,25)(17,23,24,28)(19,26,21,27);(5,9,10,8)(6,11,7,12)(14,21)(17,24)(18,19)(20,27)(23,28)(25,26)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257086, #72257130, #72257184, …
maximal normal subgroup of	#72257086, #72257130, #72257184, …
maximal quotient of	#72308857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583825);