#93794

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	C₂⁵	D₈
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

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

normal subgroups only

This group has 348 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fh,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, 990);