#93824

order	512
nilpotency class	5
isoclinism	isoclinic class of #93770
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×C₂²⋊C₈	C₂×C₄
Frattini, cosocle	#1779	C₂²
prime radical	#10320	C₂
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1779 ◁ G
LECS	G ▷ #1779 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,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, 1020);