#470884

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16706
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

696 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:

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

Polycyclic presentation:

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

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, 378080);