#406063

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14923
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

849 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)(3,13,16,23)(4,15,18,22)(5,19,21,9)(6,10)(7,11)(8,12)(14,28)(17,29)(20,26,24,25)(27,31)(30,32);(1,3)(2,4,12,18)(5,14)(6,15)(7,22)(8,16)(9,17,19,32)(10,13,11,23)(20,29)(21,31)(24,30)(25,27,26,28);(1,4)(2,3)(5,17)(6,13)(7,23)(8,18)(9,14)(10,15)(11,22)(12,16)(19,31)(20,27)(21,32)(24,28)(25,29)(26,30);(1,5,6,20,8,21,7,24)(2,9,10,25,12,19,11,26)(3,14,15,29,16,31,22,30)(4,17,13,27,18,32,23,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,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, 313259);