#391580

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14643
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

676 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:

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

Polycyclic presentation:

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

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, 298776);