#6444795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257058, #72257119, #72257173, …
maximal normal subgroup of	#72257058, #72257119, #72257173, …
maximal quotient of	#72264942

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351991);