#93820

order	512
nilpotency class	5
isoclinism	isoclinic class of #93812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1345
commutator	C₂×C₄⋊C₈	C₂×C₄
Frattini, cosocle	#1831	C₂²
prime radical	C₂³	#327
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1831 ◁ G
LECS	G ▷ #1831 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 155 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1016);