#345330

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

661 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

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

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, 252526);