#414064

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15161
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=gh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=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, 321260);