#6577830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

2188 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)(7,11)(8,12)(9,10)(14,18)(19,21)(20,22)(27,28);(6,10)(11,12)(13,14)(15,22)(16,18)(17,19)(20,24)(21,26)(23,28)(25,27);(2,4)(14,19)(15,23)(17,26)(18,21)(24,25)(27,28);(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,24)(17,23)(18,22)(19,27)(21,28)(25,26);(5,8,9,7)(6,12,10,11)(14,21)(15,23)(17,26)(18,19)(20,22)(24,25)

Polycyclic presentation:

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

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307072, #72307082, #72307670
maximal normal subgroup of	#72307072, #72307082, #72307670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485026);