#93055

order	512
nilpotency class	4
isoclinism	isoclinic class of #93047
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1294
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#431	C₂²
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

265 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=i,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gi,e^c=ehi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 251);