#6437427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6406 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305330, #72305342
maximal normal subgroup of	#72305330, #72305342
maximal quotient of	#72305619, #72305643, #72305728, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344623);