#6443100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5420 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)(9,10)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24);(5,6)(7,8)(9,11,13,19,14,20,12,21)(10,15,16,18,17,24,22,23);(1,2,3,4)(6,8);(6,8);(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312821, #72313110
maximal normal subgroup of	#72312821, #72313110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350296);