#433228

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15475
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

889 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340424);