#448851

order	512
nilpotency class	3
isoclinism	isoclinic class of #448851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15922
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356047);