#407051

order	512
nilpotency class	3
isoclinism	isoclinic class of #407051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14946
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314247);