#6439863

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

2335 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256919, #72256945, #72257035, …
maximal normal subgroup of	#72256919, #72256945, #72257035, …
maximal quotient of	#72264377, #72264411

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347059);