#427385

order	512
nilpotency class	3
isoclinism	isoclinic class of #427385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15369
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1280 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)(3,11,13,7)(4,15,17,30)(5,10)(6,9)(8,18,16,31)(12,25)(14,24,28,22)(19,20)(21,27)(23,32)(26,29);(1,2)(3,7)(4,16)(5,19)(6,9)(8,17)(10,20)(11,13)(12,21)(14,24)(15,18)(22,28)(23,26)(25,27)(29,32)(30,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,21)(9,11)(10,22)(15,29)(16,23)(17,26)(18,27)(19,24)(20,28)(25,30)(31,32);(1,4)(2,8,9,16)(3,12)(5,18)(6,17)(7,21,11,23)(10,15,19,30)(13,26)(14,27)(20,31)(22,29,24,25)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334581);