#6436540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3420 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)(6,10)(7,8)(11,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(5,6,9,11)(7,10,8,12)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(2,4)(5,7,9,8)(6,10,11,12)(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28);(5,7,9,8)(6,10,11,12)(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,25,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315557, #72315600
maximal normal subgroup of	#72315557, #72315600
maximal quotient of	#72342008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343736);