#383551

order	512
nilpotency class	3
isoclinism	isoclinic class of #383548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14361
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290747);