#93831

order	512
nilpotency class	5
isoclinism	isoclinic class of #93782
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	#1954	C₂²
prime radical	#10254	C₂
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

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

normal subgroups only

This group has 213 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)(19,26,27,22)(20,24,30,23)(21,25)(28,32,29,31);(2,7)(3,11)(4,5)(8,9)(10,16)(12,13)(14,15)(17,19)(18,22)(20,29)(21,28)(23,26)(24,32)(25,31)(27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1027);