#368553

order	512
nilpotency class	3
isoclinism	isoclinic class of #368551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14020
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275749);