#491537

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 624 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(18,22)(19,23)(20,26)(24,32)(25,30)(27,31)(28,29);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16);(3,11)(7,15)(10,16)(12,13)(17,18,23,29,20,24,27,30,21,25,31,32,26,28,19,22);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210416
maximal normal subgroup of	#72210416
maximal quotient of	#72210384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398733);