#6443145

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

5344 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315252, #72315261, #72315265
maximal normal subgroup of	#72315252, #72315261, #72315265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350341);