#455563

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	#16266
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1139 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)(3,15)(4,21)(5,24)(6,13)(7,12)(8,14)(9,18)(10,23)(11,25)(16,30)(17,32)(19,28)(20,29)(22,31)(26,27);(1,3,6,19,8,18,7,20)(2,9,12,28,14,15,13,29)(4,22,23,32,11,27,24,16)(5,17,25,31,10,30,21,26);(1,4)(2,10)(3,16)(5,14)(6,24)(7,23)(8,11)(9,26)(12,25)(13,21)(15,31)(17,29)(18,32)(19,27)(20,22)(28,30);(1,5)(2,11)(3,17)(4,14)(6,25)(7,21)(8,10)(9,27)(12,24)(13,23)(15,22)(16,28)(18,30)(19,31)(20,26)(29,32)

Polycyclic presentation:

2^9|b^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^c=gi,g^d=gi,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, 362759);