#325561

order	512
nilpotency class	3
isoclinism	isoclinic class of #278011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

676 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 170 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)(3,7,4,6)(17,18,20,22)(19,25,27,32)(21,29,26,24)(23,31,30,28);(1,3,5,4)(2,6,8,7)(9,10,11,14,12,15,16,13)(18,22)(23,30)(24,29)(25,32);(1,3,5,4)(2,7,8,6)(10,13)(11,16)(14,15)(17,19,26,31,20,27,21,28)(18,23,29,32,22,30,24,25);(1,4,5,3)(2,7,8,6)(18,22)(23,30)(24,29)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232757);