#6467083

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2300 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)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(15,21)(16,22)(18,24)(20,25)(23,27)(26,28);(6,9)(10,12)(13,15,22,27,17,23,16,21)(14,18,25,28,19,26,20,24);(13,15,22,27,17,23,16,21)(14,18,25,28,19,26,20,24);(5,7,8,11)(6,10,9,12)(14,19)(18,26)(20,25)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321749, #72321812, #72321820
maximal normal subgroup of	#72321749, #72321812, #72321820

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374279);