#6437949

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2410
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,6,10,8,12,7,11)(3,13,17,28,18,27,16,9)(4,19,21,25,23,26,22,5)(14,15,20,24,30,32,29,31);(1,3)(2,9)(4,20)(5,24)(6,16)(7,17)(8,18)(10,27)(11,13)(12,28)(14,21)(15,19)(22,30)(23,29)(25,31)(26,32);(1,4)(2,5)(3,14)(6,21)(7,22)(8,23)(9,15)(10,19)(11,26)(12,25)(13,24)(16,29)(17,20)(18,30)(27,31)(28,32);(1,5)(2,4)(3,15)(6,19)(7,26)(8,25)(9,14)(10,21)(11,22)(12,23)(13,20)(16,31)(17,24)(18,32)(27,29)(28,30);(1,6,8,7)(2,10,12,11)(3,16,18,17)(4,21,23,22)(5,19,25,26)(9,27,28,13)(14,29,30,20)(15,31,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214082, #72215784, #72215974, …
maximal normal subgroup of	#72214082, #72215784, #72215974, …
maximal quotient of	C₄×#24472, C₂×#6437949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345145);