#6443173

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	#34936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5178 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)(9,10,15,12)(11,19,21,20)(13,18,14,17)(16,23,24,22);(1,2,3,4)(9,11,13,22,15,21,14,23)(10,16,17,19,12,24,18,20);(5,6,7,8)(9,10,15,12)(11,19,21,20)(13,18,14,17)(16,23,24,22);(9,12,15,10)(11,20,21,19)(13,17,14,18)(16,22,24,23);(11,21)(16,24)(19,20)(22,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^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,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315219, #72315265
maximal normal subgroup of	#72315219, #72315265
maximal quotient of	#72315733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350369);