#304745

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=efh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211941);