#491201

order	512
nilpotency class	4
isoclinism	isoclinic class of #491097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18,21,24)(19,26,27,31)(20,29,30,32)(22,28,23,25);(2,7)(8,15)(9,10)(11,16)(17,19,26,24,22,29,20,28,21,27,31,18,23,32,30,25);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(18,24)(19,27)(25,28)(29,32);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)

Polycyclic presentation:

2^9|a^2=i,b^2=eghi,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398397);