#330523

order	512
nilpotency class	3
isoclinism	isoclinic class of #329551
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

587 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 256

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

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

Polycyclic presentation:

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

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, 237719);