#397512

order	512
nilpotency class	3
isoclinism	isoclinic class of #397512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1243 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304708);