#6448819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3440 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)(8,9)(11,12)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(2,4)(13,14)(15,23)(16,22)(17,19)(18,21)(20,25)(24,27)(26,28);(6,9)(10,12)(13,15,18,26)(14,20,21,27)(16,25,17,24)(19,23,22,28);(5,7,8,11)(6,10,9,12)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(14,21)(19,22)(20,27)(23,28)

Polycyclic presentation:

2^9|c^2=i,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315249, #72315268, #72315279
maximal normal subgroup of	#72315249, #72315268, #72315279
maximal quotient of	#72316322, #72316324

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356015);