#325375

order	512
nilpotency class	3
isoclinism	isoclinic class of #278155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32);(1,3,5,4)(2,6,8,7)(10,14)(13,16)(17,19,26,27,20,28,21,29)(18,22,30,32,23,25,24,31);(1,3,5,4)(2,7,8,6)(9,10,11,13,12,14,15,16)(18,23)(19,27)(21,26)(22,31)(25,32)(28,29);(1,4,5,3)(2,7,8,6)(10,14)(13,16)(18,23)(22,25)(24,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232571);