#93833

order	512
nilpotency class	5
isoclinism	isoclinic class of #93790
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1344
commutator	C₂×C₄×C₈	C₂×C₄
Frattini, cosocle	#1954	C₂²
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1954 ◁ G
LECS	G ▷ #1954 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

145 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 30 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=fh,d^2=hi,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=defhi,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, 1029);