#6444388

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2946 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)(9,10,13,18,15,17,14,12)(11,19,21,20,23,24,22,16);(5,6)(7,8)(9,11,15,23)(10,16,17,20)(12,19,18,24)(13,22,14,21);(1,2,3,4)(6,8)(9,12,14,17,15,18,13,10)(11,20,22,19,23,16,21,24);(6,8)(9,10,13,18,15,17,14,12)(11,19,21,20,23,24,22,16);(10,17)(12,18)(16,20)(19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307351, #72307626, #72315259, …
maximal normal subgroup of	#72307351, #72307626, #72315259, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351584);