#425251

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15331
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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)(3,14,18,29)(4,9)(5,22)(6,11)(7,12)(8,17,28,19)(10,24)(13,26)(15,30,20,23)(16,27,32,31)(21,25);(1,3)(2,8)(4,20)(5,23)(6,19)(7,18)(9,16)(10,15)(11,29)(12,28)(13,14)(17,26)(21,31)(22,27)(24,32)(25,30);(1,4)(2,9)(3,15,18,20)(5,13)(6,21)(7,10)(8,16,28,32)(11,25)(12,24)(14,23,29,30)(17,31,19,27)(22,26);(1,5,6,9,7,25,26,24)(2,10,11,22,12,4,13,21)(3,16,17,23,18,32,19,30)(8,27,14,15,28,31,29,20)

Polycyclic presentation:

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

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, 332447);