#471108

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16754
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,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, 378304);