#6583870

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2425 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)(7,12)(8,10)(9,11)(13,14)(15,18)(16,19)(17,20)(21,26)(22,27)(23,25)(24,28);(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,27)(20,26)(23,24)(25,28);(1,2,3,4)(14,19)(15,21)(17,24)(18,25)(22,23)(26,27);(5,8,9,7)(6,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,27)(20,26)(23,24)(25,28);(15,21)(17,24)(18,26)(20,28)(22,23)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257277, #72257414, #72263856
maximal normal subgroup of	#72257277, #72257414, #72263856
maximal quotient of	#72264246

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491066);