#433019

order	512
nilpotency class	3
isoclinism	isoclinic class of #433019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 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:

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

Polycyclic presentation:

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

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, 340215);