#458074

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16309
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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)(3,12)(4,9)(5,18)(6,11)(7,14)(8,17)(10,20)(13,28)(15,27)(16,24)(19,25)(21,29)(22,32)(23,31)(26,30);(1,3,5,14,6,15,20,32)(2,7,10,12,11,22,18,27)(4,13,19,31,17,30,16,29)(8,23,25,28,9,21,24,26);(1,4,5,19,6,17,20,16)(2,8,10,25,11,9,18,24)(3,13,14,31,15,30,32,29)(7,21,12,26,22,23,27,28);(1,4)(2,9)(3,13)(5,19)(6,17)(7,21)(8,11)(10,24)(12,26)(14,31)(15,30)(16,20)(18,25)(22,23)(27,28)(29,32)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 365270);