#471046

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16739
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378242);