#448912

order	512
nilpotency class	3
isoclinism	isoclinic class of #448911
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15925
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

699 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=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, 356108);