#414301

order	512
nilpotency class	3
isoclinism	isoclinic class of #412862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2858
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15167
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

991 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,21)(14,26)(16,25)(17,24)(19,32)(20,28)(22,27)(23,31)(29,30);(1,3)(2,8)(4,19,21,31)(5,16)(6,15)(7,17)(9,26,18,29)(10,13)(11,24)(12,25)(14,22,30,20)(23,28,32,27);(1,4,5,20,7,21,6,22)(2,9,11,28,12,18,10,27)(3,14,15,19,17,30,16,31)(8,23,13,29,25,32,24,26);(2,10,12,11)(8,13,25,24)(9,27,18,28)(23,29,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321497);