#6352606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352581
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260479
maximal normal subgroup of	#72260479
maximal quotient of	#72261762, #72261773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259802);