#234223

order	512
nilpotency class	3
isoclinism	isoclinic class of #234214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1363 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141419);