#427384

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15369
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1224 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)(3,8)(4,17,19,9)(5,13)(6,12)(7,11)(10,15)(14,29,30,23)(16,24)(18,26,31,32)(20,27,22,28)(21,25);(1,3)(2,8)(4,18)(5,21)(6,15)(7,16)(9,26)(10,12)(11,24)(13,25)(14,20)(17,32)(19,31)(22,30)(23,27)(28,29);(1,4)(2,9)(3,14)(5,22)(6,20)(7,19)(8,23)(10,26)(11,17)(12,28)(13,27)(15,31)(16,30)(18,21)(24,29)(25,32);(1,3)(2,10,11,25)(4,14)(5,21)(6,15)(7,16)(8,12,24,13)(9,26,17,32)(18,22)(19,30)(20,31)(23,28,29,27)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334580);