#6671463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2333 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)(6,9)(7,11);(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,20,17,19)(16,24,25,27)(21,26,28,23);(2,4)(14,19)(15,17)(20,22)(21,24)(23,26)(27,28);(5,7,8,11)(6,10,9,12)(13,15,18,17)(14,20,22,19)(16,23,25,26)(21,27,28,24);(6,9)(10,12)(13,16,18,25)(14,21,22,28)(15,23,17,26)(19,24,20,27)

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,e^2=i,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	#72257103, #72257139, #72257173
maximal normal subgroup of	#72257103, #72257139, #72257173
maximal quotient of	#72308915

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578659);