#6477301

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469932
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3480 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)(6,9)(7,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(5,6,8,10)(7,12,11,9)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(2,4)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27);(15,22)(18,26)(21,23)(25,27);(14,20)(18,26)(19,28)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315541, #72315600
maximal normal subgroup of	#72315541, #72315600
maximal quotient of	#72316144, #72316244

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384497);