#6348999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
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	C₄²	C₂⁵
prime radical	#41²	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271558, #72271563, #72271566, …
maximal normal subgroup of	#72271558, #72271563, #72271566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256195);