#6347465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347429
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35165	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257350, #72257370, #72257447
maximal normal subgroup of	#72257350, #72257370, #72257447
maximal quotient of	#72264441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254661);