#93778

order	512
nilpotency class	5
isoclinism	isoclinic class of #93776
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₄×C₈	C₂×C₄
Frattini, cosocle	#1984	C₂²
prime radical	C₂×#493	C₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1984 ◁ G
LECS	G ▷ #1984 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

235 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 32 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fhi,d^2=i,e^2=h,f^2=hi,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,d^c=dgh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 974);