#464967

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16512
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 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)(3,12)(4,18)(5,21)(6,23)(7,11)(8,15)(9,20)(10,22)(13,25)(14,24)(16,26)(17,27)(19,28)(29,32)(30,31);(1,3)(2,8,11,26)(4,13)(5,14)(6,15)(7,17)(9,25,28,24)(10,12,23,27)(16,22)(18,30,21,29)(19,31)(20,32);(1,4,6,20,7,5,22,19)(2,9,10,18,11,28,23,21)(3,13,15,32,17,14,16,31)(8,24,12,29,26,25,27,30);(1,5)(2,9)(3,14)(4,7)(6,19)(8,25)(10,18)(11,28)(12,30)(13,17)(15,31)(16,32)(20,22)(21,23)(24,26)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372163);