#420378

order	512
nilpotency class	3
isoclinism	isoclinic class of #420377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15277
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1321 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338621, #72338623, #72338629
maximal normal subgroup of	#72338621, #72338623, #72338629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327574);